Solved on Feb 26, 2024

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

STEP 1

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

STEP 2

To factor the quadratic equation, we need to find two numbers that multiply to give the product of the coefficient of $x^2$ (which is 1 in this case) and the constant term (which is 18) and also add up to give the coefficient of $x$ (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 $x$ is negative.
$\begin{align} 1 \times 18 &= 18 \\ 2 \times 9 &= 18 \\ 3 \times 6 &= 18 \\ (-1) \times (-18) &= 18 \\ (-2) \times (-9) &= 18 \\ (-3) \times (-6) &= 18 \\ \end{align}$

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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

STEP 1

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

STEP 2

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

STEP 3

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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Solve the quadratic equation x2−9x+18=0x^{2} - 9x + 18 = 0x2−9x+18=0 by factoring.

STEP 1

Assumptions1. We are given a quadratic equation in the form ax2+bx+c=0 ax^2 + bx + c = 0 ax2+bx+c=0.2. We need to solve for x x x by factoring the quadratic equation x2−9x+18=0 x^2 - 9x + 18 = 0 x2−9x+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 x2 (which is 1 in this case) and the constant term (which is 18) and also add up to give the coefficient of x x x (which is -9).

STEP 3

STEP 4

From the list of factor pairs, identify the pair that adds up to -9 (the coefficient of x x x).(−3)+(−6)=−9 (-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.x2−9x+18=(x−3)(x−6) x^2 - 9x + 18 = (x - 3)(x - 6) x2−9x+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 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.(x−3)(x−6)=0 (x - 3)(x - 6) = 0 (x−3)(x−6)=0

STEP 7

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

STEP 8

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

STEP 9

Solve the second equation for x x x.x−6=0⇒x=6 x - 6 = 0 \Rightarrow x = 6 x−6=0⇒x=6The solutions to the quadratic equation x2−9x+18=0 x^2 - 9x + 18 = 0 x2−9x+18=0 are x=3 x = 3 x=3 and x=6 x = 6 x=6.

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

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

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

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

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

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

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

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

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