PSAT Math - How to find the solution to a quadratic equation

Solve 3x2 + 10x = –3

x = –1/3 or –3

x = –1/6 or –6

x = –1/9 or –9

x = –2/3 or –2

x = –4/3 or –1

Explanation

Generally, quadratic equations have two answers.

First, the equations must be put in standard form: 3x2 + 10x + 3 = 0

Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.

Third, check the answer by plugging the answers back into the original equation.

If $x(x^2-5x+3)=-9, \ x>0,\ and \ y^2+y=2$ then which of the following is a possible value for ?

Explanation

$x(x^2-5x+3)=-9, \ x>0,\ and \ y^2+y=2$

$x(x^2-5x+3)=-9 \Rightarrow x^3-5x^2+3x + 9=0\Rightarrow (x^2-6x+9)(x+1)$

$(x^2-6x+9)(x+1) = (x-3)^2(x+1)\Rightarrow x= -1\ or\ 3$

Since , .

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

Thus $xy^2+y = (3)(-2)^2+(-2)\ or\ (3)(1)^2+(1) = 12-2\ or\ 3+1\ = 10\ or\ 4$

Of these two, only 4 is a possible answer.

The expression $x^{2} - 8x +12$ is equal to 0 when $x = 2$ and $x = ?$

$6$

$-12$

$-6$

$-2$

$4$

Explanation

Factor the expression and set each factor equal to 0:

$(x-2)(x-6)= 0$

$x-2 = 0$

$x = 2$

$x-6 = 0$

$x = 6$

Let f(x) = 2_x_2 – 4_x_ + 1 and g(x) = (_x_2 + 16)(1/2). If k is a negative number such that f(k) = 31, then what is the value of (f(g(k))?

-81

-35

Explanation

In order to find the value of f(g(k)), we will first need to find k. We are told that f(k) = 31, so we can write an expression for f(k) and solve for k.

f(x) = 2_x_2 – 4_x_ + 1

f(k) = 2_k_2 – 4_k_ + 1 = 31

Subtract 31 from both sides.

2_k_2 – 4_k –_ 30 = 0

Divide both sides by 2.

k_2 – 2_k – 15 = 0

Now, we can factor this by thinking of two numbers that multiply to give –15 and add to give –2. These two numbers are –5 and 3.

k_2 –2_k – 15 = (k – 5)(k + 3) = 0

We can set each factor equal to 0 to find the values for k.

k – 5 = 0

Add 5 to both sides.

k = 5

Now we set k + 3 = 0.

Subtract 3 from both sides.

k = –3

This means that k could be either 5 or –3. However, we are told that k is a negative number, which means k = –3.

Finally, we can evaluate the expression f(g(–3)). First we need to find g(–3).

g(x) = (_x_2 + 16)(1/2)

g(–3) = ((–3)2 + 16)(1/2)

= (9 + 16)(1/2)

= 25(1/2)

Raising something to the one-half power is the same as taking the square root.

25(1/2) = 5

Now that we know g(–3) = 5, we must find f(5).

f(5) = 2(5)2 – 4(5) + 1

= 2(25) – 20 + 1 = 31

The answer is 31.

A rectangle has a perimeter of $50\ m$ and an area of $150\ m^{2}$ What is the difference between the length and width?

$5\ m$

$10\ m$

$15\ m$

$20\ m$

$25\ m$

Explanation

For a rectangle, $P=2w+2l$ and $A=lw$ where $w$ = width and $l$ = length.

So we get two equations with two unknowns:

$50=2w+2l$

$25=w+l$

$l=25-w$

$150=lw$

Making a substitution we get

$150=(25-w)w$

$w^{2} -25w + 150 = 0$

Solving the quadratic equation we get $w=10\ m$ or $15\ m$ .

$l=15\ m\ or\ 10\ m$

The difference is $5\ m$ .

Two consecutive positive numbers have a product of 420. What is the sum of the two numbers?

Explanation

Let = first positive number and = second positive number

So the equation to solve becomes

Using the distributive property, multiply out the equation and then set it equal to 0. Next factor to solve the quadratic.

Find all real solutions to the equation.

$x{^{2}+6x-16=0$

$2\ \text{or}\ -8$

$-2\ \text{or}\ -8$

$4\ \text{or}\ -4$

Explanation

$x{^{2}+6x-16=0$

To solve by factoring, we need two numbers that add to $\small 6$ and multiply to $\small -16$ .

$-2*8=-16\ \text{and}\ -2+8=6$

In order for the equation to equal zero, one of the terms must be equal to zero.

Our final answer is that $x\epsilon\left { 2,-8 \right }$ .

Solve for :

$4x^{2} + 9=3(x^{2}+4)$

$\pm \sqrt{3}$

$\frac{3}{4}$

$\frac{3}{2}$

Explanation

Begin by distributing the three on the right side of the equation: $4x^{2}+9=3x^{2}+12$

Next combine your like terms by subtracting $3x^{2}$ from both sides to give you $x^{2}+9=12$

Next, subtract 9 from both sides to give you $x^{2}=3$ . To solve for , now take the square root of both sides. This gives you the answer, $x=\pm \sqrt{3}$