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Solution - Solving quadratic equations by factoring

Exact form:

x = 9

x=9

Decimal form:

x = 9

x=9

Equation in factored form:

{(x - 9)}^{2} = 0

(x-9)^2=0

See steps

Step-by-step explanation

1. Make sure that the equation is a perfect square trinomial

In a perfect square trinomial, the rule is that the square root of coefficient $a$ times the square root of coefficient $c$ times two equals coefficient $b$ :
$\sqrt{a} \cdot \sqrt{c} \cdot 2 = b$

To find the coefficients, use the standard form of a quadratic equation:
$a x^{2} + b x + c = 0$
$x^{2} - 18 x + 81 = 0$

Coefficient $a$ $= 1$
Coefficient $b$ $= - 18$
Coefficient $c$ $= 81$

Plug the coefficients into the rule and check if it's true:

$\sqrt{(1)} \cdot \sqrt{(81)} \cdot 2 = - 18$

Take out the square roots

$1 \cdot 9 \cdot 2 = - 18$

Simplify the expression

$18 = - 18$

Because the equation is true,
$x^{2} - 18 x + 81$
is a perfect square trinomial.

2. Find the factor of the perfect square trinomial

To find the factor of the perfect square trinomial:
$x^{2} - 18 x + 81$
Use the perfect square trinomial formula:
$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$

$a^{2} = x^{2}$

$b^{2} = 81$

Take out the square roots

$a = \sqrt{x^{2}}$

$b = \sqrt{81}$

Simplify the expression

$a = x$

$b = 9$

${(a + b)}^{2} = {(x - 9)}^{2}$
The factor of $x^{2} - 18 x + 81$ is $(x - 9)$

3. Find the root of the quadratic equation

Find the root of:
$x^{2} - 18 x + 81 = 0$
by using its factored form:
${(x - 9)}^{2} = 0$

If
${(x - 9)}^{2} = 0$
Then
$(x - 9) (x - 9) = 0$
Which means
$(x - 9) = 0$

Solve for $x$ :

2 additional steps

$(x - 9) = 0$

Add to both sides:

$(x - 9) + 9 = 0 + 9$

Simplify the arithmetic:

$x = 0 + 9$

Simplify the arithmetic:

$x = 9$

4. Graph

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Why learn this

In their most basic function, quadratic equations define shapes like circles, ellipses and parabolas. These shapes can, in turn, be used to predict the curve of an object in motion, such as a ball kicked by a football player or a shot fired out of a cannon.
When it comes to an object’s movement through space, what better place to start than space itself, with the revolution of planets around the sun in our solar system? The quadratic equation was used to establish that planets’ orbits are elliptical, not circular. Determining the path and speed an object travels through space is possible even after it has come to a stop: the quadratic equation can calculate how fast a vehicle was moving when it crashed. With information like this, the automotive industry can design brakes to prevent collisions in the future. Many industries use the quadratic equation to predict and thus improve their products’ lifespan and safety.

Terms and topics

Solving quadratic equations by factoring