Math

Question Solve the quadratic equation x29x+18=0x^{2} - 9x + 18 = 0 by factoring.

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form ax2+bx+c=0 ax^2 + bx + c = 0 .
2. We need to solve for x x by factoring the quadratic equation x29x+18=0 x^2 - 9x + 18 = 0 .

STEP 2

To factor the quadratic equation, we need to find two numbers that multiply to give the product of the coefficient of x2 x^2 (which is 1 in this case) and the constant term (which is 18) and also add up to give the coefficient of x x (which is -9).

STEP 3

List the pairs of factors of 18 (the constant term) that could multiply to 18, considering both positive and negative factors because the coefficient of x x is negative.
1×18=182×9=183×6=18(1)×(18)=18(2)×(9)=18(3)×(6)=18 \begin{align*} 1 \times 18 &= 18 \\ 2 \times 9 &= 18 \\ 3 \times 6 &= 18 \\ (-1) \times (-18) &= 18 \\ (-2) \times (-9) &= 18 \\ (-3) \times (-6) &= 18 \\ \end{align*}

STEP 4

From the list of factor pairs, identify the pair that adds up to -9 (the coefficient of x x ).
(3)+(6)=9 (-3) + (-6) = -9

STEP 5

Use the identified pair of factors to write the quadratic equation as a product of two binomials.
x29x+18=(x3)(x6) x^2 - 9x + 18 = (x - 3)(x - 6)

STEP 6

Now that we have factored the quadratic equation, we can use the zero product property to solve for x x . The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero.
(x3)(x6)=0 (x - 3)(x - 6) = 0

STEP 7

Set each factor equal to zero and solve for x x .
x3=0orx6=0 x - 3 = 0 \quad \text{or} \quad x - 6 = 0

STEP 8

Solve the first equation for x x .
x3=0x=3 x - 3 = 0 \Rightarrow x = 3

STEP 9

Solve the second equation for x x .
x6=0x=6 x - 6 = 0 \Rightarrow x = 6
The solutions to the quadratic equation x29x+18=0 x^2 - 9x + 18 = 0 are x=3 x = 3 and x=6 x = 6 .

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