Calculate Maximum or Minimum of Quadratic by Completing the Square: CCSS.Math.Content.HSA-SSE.B.3b - Algebra

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+4x+7=0\Rightarrow (x+2)^2+2^2+7=0+2^2$

When simplified the new function is,

$\(x+2)^2+4+7=4 \(x+2)^2+11=4 \(x+2)^2=-7$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=(-2)^2+4(-2)+7 \y=4-8+7 \y=3$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+2x-4=0\Rightarrow (x+1)^2+1^2-4=0+1^2$

When simplified the new function is,

$\(x+1)^2-4+1 \(x+1)^2-3$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+1=0 \x+1-1=0-1 \x=-1$

From here, substitute the the value into the original function.

$\y=(-1)^2+2(-1)-4 \y=1-2-4 \y=-5$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 8$

Now divide the middle term coefficient by two.

$\frac{8}{2}=4$

From here write the function with the perfect square.

$x^2+8x-3\Rightarrow (x+4)^2-3+4^2$

When simplified the new function is,

$\(x+4)^2-3+16\(x+4)^2+13$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+4=0 \x+4-4=0-4 \x=-4$

From here, substitute the the value into the original function.

$\y=(-4)^2+8(-4)-3 \y=16-32-3 \y=-19$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$x^2+6x+2\Rightarrow (x+3)^2+6+3^2$

When simplified the new function is,

$\(x+3)^2+2+9\(x+3)^2+11$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+3=0 \x+3-3=0-3 \x=-3$

From here, substitute the the value into the original function.

$\y=(-3)^2+6(-3)+2 \y=9-18+2 \y=-7$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -4$

Now divide the middle term coefficient by two.

$\frac{-4}{2}=-2$

From here write the function with the perfect square.

$x^2-4x+1\Rightarrow (x-2)^2+1+(-2)^2$

When simplified the new function is,

$\(x-2)^2+1+4\(x-2)^2+5$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x-2=0 \x-2+2=0+2\x=2$

From here, substitute the the value into the original function.

$\y=(2)^2-4(2)+1 \y=4-8+1 \y=-3$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -6$

Now divide the middle term coefficient by two.

$\frac{-6}{2}=-3$

From here write the function with the perfect square.

$x^2-6x-6\Rightarrow (x-3)^2-6+(-3)^2$

When simplified the new function is,

$\(x-3)^2-6+9\(x-3)^2+3$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x-3=0 \x-3+3=0+3\x=3$

From here, substitute the the value into the original function.

$\y=(3)^2-6(3)-6 \y=9-18-6 \y=-15$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square.

$-(x^2+2x+1)\Rightarrow-((x+1)^2+1+1)$

When simplified the new function is,

$\-((x+1)^2+2)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+1=0 \x+1-1=0-1 \x=-1$

From here, substitute the the value into the original function.

$\y=-(-1)^2-2(-1)-1 \y=-1+2-1 \y=0$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+1)\Rightarrow-((x+2)^2+1+2^2)$

When simplified the new function is,

$\-((x+2)^2+5)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=-(-2)^2-4(-2)-1 \y=-4+8-1 \y=3$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$-(x^2+6x+1)\Rightarrow-((x+3)^2+1+3^2)$

When simplified the new function is,

$\-((x+3)^2+10)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+3=0 \x+3-3=0-3 \x=-3$

From here, substitute the the value into the original function.

$\y=-(-3)^2-6(-3)-1 \y=-9+18-1 \y=8$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 8$

Now divide the middle term coefficient by two.

$\frac{8}{2}=4$

From here write the function with the perfect square.

$-(x^2+8x+2)\Rightarrow-((x+4)^2+2+4^2)$

When simplified the new function is,

$\-((x+4)^2+18)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+4=0 \x+4-4=0-4 \x=-4$

From here, substitute the the value into the original function.

$\y=-(-4)^2-8(-4)-2 \y=-16+32-2 \y=14$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+5)\Rightarrow-((x+2)^2+5+2^2)$

When simplified the new function is,

$\-((x+2)^2+9)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=-(-2)^2-4(-2)-5 \y=-4+8-5 \y=-1$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$-(x^2+6x+5)\Rightarrow-((x+3)^2+5+3^2)$

When simplified the new function is,

$\-((x+3)^2+14)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+3=0 \x+3-3=0-3 \x=-3$

From here, substitute the the value into the original function.

$\y=-(-3)^2-6(-3)-5 \y=-9+18-5 \y=4$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+4x+7=0\Rightarrow (x+2)^2+2^2+7=0+2^2$

When simplified the new function is,

$\(x+2)^2+4+7=4 \(x+2)^2+11=4 \(x+2)^2=-7$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=(-2)^2+4(-2)+7 \y=4-8+7 \y=3$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+2x-4=0\Rightarrow (x+1)^2+1^2-4=0+1^2$

When simplified the new function is,

$\(x+1)^2-4+1 \(x+1)^2-3$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+1=0 \x+1-1=0-1 \x=-1$

From here, substitute the the value into the original function.

$\y=(-1)^2+2(-1)-4 \y=1-2-4 \y=-5$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 8$

Now divide the middle term coefficient by two.

$\frac{8}{2}=4$

From here write the function with the perfect square.

$x^2+8x-3\Rightarrow (x+4)^2-3+4^2$

When simplified the new function is,

$\(x+4)^2-3+16\(x+4)^2+13$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+4=0 \x+4-4=0-4 \x=-4$

From here, substitute the the value into the original function.

$\y=(-4)^2+8(-4)-3 \y=16-32-3 \y=-19$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$x^2+6x+2\Rightarrow (x+3)^2+6+3^2$

When simplified the new function is,

$\(x+3)^2+2+9\(x+3)^2+11$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+3=0 \x+3-3=0-3 \x=-3$

From here, substitute the the value into the original function.

$\y=(-3)^2+6(-3)+2 \y=9-18+2 \y=-7$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -4$

Now divide the middle term coefficient by two.

$\frac{-4}{2}=-2$

From here write the function with the perfect square.

$x^2-4x+1\Rightarrow (x-2)^2+1+(-2)^2$

When simplified the new function is,

$\(x-2)^2+1+4\(x-2)^2+5$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x-2=0 \x-2+2=0+2\x=2$

From here, substitute the the value into the original function.

$\y=(2)^2-4(2)+1 \y=4-8+1 \y=-3$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -6$

Now divide the middle term coefficient by two.

$\frac{-6}{2}=-3$

From here write the function with the perfect square.

$x^2-6x-6\Rightarrow (x-3)^2-6+(-3)^2$

When simplified the new function is,

$\(x-3)^2-6+9\(x-3)^2+3$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x-3=0 \x-3+3=0+3\x=3$

From here, substitute the the value into the original function.

$\y=(3)^2-6(3)-6 \y=9-18-6 \y=-15$

Therefore the minimum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square.

$-(x^2+2x+1)\Rightarrow-((x+1)^2+1+1)$

When simplified the new function is,

$\-((x+1)^2+2)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+1=0 \x+1-1=0-1 \x=-1$

From here, substitute the the value into the original function.

$\y=-(-1)^2-2(-1)-1 \y=-1+2-1 \y=0$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+1)\Rightarrow-((x+2)^2+1+2^2)$

When simplified the new function is,

$\-((x+2)^2+5)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=-(-2)^2-4(-2)-1 \y=-4+8-1 \y=3$

Therefore the maximum value occurs at the point .

Compare your answer with the correct one above

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