ACT Math - How to find the solution to a quadratic equation a1

Solve for :

Round to the nearest hundredth.

$-6\textup{ and}-3$

$\textup{No solution}$

$6\textup{ and }3$

$-4.44\textup{ and }2.41$

$4.44\textup{ and}-2.41$

Explanation

With quadratic equations, you should always start by getting all of your terms to one side of the equation, setting this equal to :

Thus, simplify into:

Now, the next question you need to ask yourself is, "Can this be factored?" In this case, it can. Factor the quadratic expression:

Now, remember that you merely need to set each group equal to . This gives you the two values for :

; therefore

Likewise, for the other group,

Solve for :

$-6\textup{ and }\frac{-1}{3}$

$\textup{No solution}$

$3\textup{ and }6$

$\frac{2}{5}\textup{ and }\frac{4}{5}$

$\frac{1 + \sqrt{6}}{3}\textup{ and }\frac{1 - \sqrt{6}}{3}$

Explanation

With quadratic equations, you should always start by getting all of your terms to one side of the equation, setting this equal to :

Thus, simplify into:

Now, the next question you need to ask yourself is, "Can this be factored?" In this case, it can, though we are sometimes a bit intimidated by terms that have a coefficient like this. Factor the quadratic expression:

If you FOIL this out, you will see your original equation.

Now, remember that you merely need to set each group equal to . This gives you the two values for :

$x=\frac{-1}{3}$

For the other group, you get .

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

Explanation

Let be defined as the lower number, and as the greater number.

We know that the first number times the second is , so the equation to solve becomes .

Distributing the gives us a polynomial, which we can solve by factoring.

$x^{2} + 3x - 180 = 0$

and

The question tells us that the integers are positive; therefore, .

If , and the second number is , then the second number is .

The sum of these numbers is .

Solve for x: (x2 – x) / (x – 1) = 1

No solution

x = 1

x = -1

x = 2

x = -2

Explanation

Begin by multiplying both sides by (x – 1):

x2 – x = x – 1

Solve as a quadratic equation: x2 – 2x + 1 = 0

Factor the left: (x – 1)(x – 1) = 0

Therefore, x = 1.

However, notice that in the original equation, a value of 1 for x would place a 0 in the denominator. Therefore, there is no solution.

Solve for :

$\textup{No real solution}$

$\frac{1 \pm \sqrt{491}}{-2}$

$-12\textup{ and }4$

$\frac{-5 \pm \sqrt{31}}{3}$

$-8.5\textup{ and}-3.5$

Explanation

With quadratic equations, you should always start by getting all of your terms to one side of the equation, setting this equal to :

Thus, simplify into:

Now, the next question you need to ask yourself is, "Can this be factored?" In this case, it cannot be easily factored. Therefore, you should use the quadratic formula. Recall that its general form is:

$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

For our data, , , and .

Thus, we have:

$\frac{-(-1)\pm\sqrt{(-1)^2-(4*7*35)}}{2*-1}$

Simplifying, this is:

$\frac{1\pm\sqrt{1-(980)}}{-2}$

Since is negative, you know that there is no real solution (given the problems arising from the negative square root)!

The height of a ball (in feet) after it is thrown in the air is given by the expression

s(t) = –t_2 + 4_t

where t is time in seconds. The ball is thrown from ground level at t = 0. How many seconds will pass before the ball reaches the ground again?

Explanation

Notice that when the ball is at ground level, the height is zero. Setting s (t) equal to zero and solving for t will then give the times when the ball is at the ground.

–t_2 + 4_t =0

t(4 – t) = 0

t = 0, t = 4

The ball returns to the ground after 4 seconds.

Two positive consecutive multiples of three have a product of 108. What is their sum?