Question Find all values of $x$ that solve the equation $2x^2 - 13x + 36 = 15$ .

Studdy Solution

STEP 1

Assumptions
1. The function given is $f(x) = 2x^2 - 13x + 36$ .
2. We need to find the values of $x$ such that $f(x) = 15$ .

STEP 2

To find the values of $x$ for which $f(x) = 15$ , we set the function equal to 15 and solve for $x$ .
$2x^2 - 13x + 36 = 15$

STEP 3

Subtract 15 from both sides of the equation to set the right side to zero, which is necessary for solving a quadratic equation.
$2x^2 - 13x + 36 - 15 = 0$

STEP 4

Simplify the equation by combining like terms.
$2x^2 - 13x + 21 = 0$

STEP 5

We now have a quadratic equation in standard form. We can try to factor this equation, or use the quadratic formula to find the values of $x$ . Let's attempt to factor first.

STEP 6

Look for two numbers that multiply to $2 \times 21$ (which is 42) and add up to -13 (the coefficient of the $x$ term).

STEP 7

The numbers that satisfy these conditions are -6 and -7 because $(-6) \times (-7) = 42$ and $(-6) + (-7) = -13$ .

STEP 8

Rewrite the middle term of the quadratic equation using the numbers found in STEP_7.
$2x^2 - 6x - 7x + 21 = 0$

STEP 9

Factor by grouping. Group the first two terms together and the last two terms together.
$(2x^2 - 6x) + (-7x + 21) = 0$

STEP 10

Factor out the greatest common factor from each group.
$2x(x - 3) - 7(x - 3) = 0$

STEP 11

Now, factor out the common binomial factor $(x - 3)$ .
$(2x - 7)(x - 3) = 0$

STEP 12

Apply the zero-product property, which states that if a product of two factors is zero, then at least one of the factors must be zero.
$2x - 7 = 0 \quad \text{or} \quad x - 3 = 0$

STEP 13

Solve each equation for $x$ .
First, solve $2x - 7 = 0$ :
$2x = 7$
$x = \frac{7}{2}$
$x = 3.5$

STEP 14

Now, solve $x - 3 = 0$ :
$x = 3$

STEP 15

The solutions to the equation $f(x) = 15$ are $x = 3.5$ and $x = 3$ .
Karan needs to pay $803 each month.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms