Problem 14 Solve each equation. $$ x^{2... [FREE SOLUTION]

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Chapter 5: Problem 14

Solve each equation. $$ x^{2}-18 x+80=0 $$

Short Answer

Expert verified

The solutions are $x = 10$ and $x = 8$.

Step by step solution

Identify the equation type

The given equation is a quadratic equation of the form $x^{2} - 18x + 80 = 0$.

Factor the quadratic equation

To factor the quadratic equation, look for two numbers that multiply to the constant term (+80) and add up to the linear coefficient (-18). These numbers are -10 and -8. Therefore, the equation can be factored as: $x^{2} - 18x + 80 = (x-10)(x-8)$.

Set each factor to zero

Set each factor equal to zero to find the values of $x$: $x-10 = 0$ $x-8 = 0$

Solve for $x$

Solve each equation separately to get the solutions for $x$: $x-10=0$ $x=10$ $x-8=0$ $x=8$ So, the solutions are $x = 10$ and $x = 8$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Quadratic Equations

Factoring quadratic equations is an essential method for solving them. In a quadratic equation like $x^2 - 18x + 80 = 0$, we need to express it as a product of two binomials of the form $(x-a)(x-b)$. To do this, let's find two numbers that multiply to the constant term (80) and add to the coefficient of the linear term (-18). Here, the suitable numbers are -10 and -8. We then rewrite the equation as $(x-10)(x-8)$. This technique transforms the equation into a simpler form, making it easier to solve.

Solving Quadratic Equations

Once the quadratic equation is factored, solving becomes straightforward. With our equation $(x-10)(x-8) = 0$, we now have two linear equations. These equations are much simpler to handle. The solutions to these linear equations will provide the solutions to our original quadratic equation. Always remember: factoring a quadratic equation reduces it to simpler components that are easier to solve.

Setting Factors to Zero

After factoring the quadratic equation, we set each factor to zero because if the product of two factors is zero, at least one of them must be zero. This principle is based on the Zero Product Property. For the equation $(x-10)(x-8) = 0$, we set each factor to zero individually:

$x-10 = 0$
$x-8 = 0$

By doing this, we create two simpler equations that we can solve to find our values for $x$. This step is crucial in finding the actual solutions of the quadratic equation.

Solutions to Quadratic Equations

The final step in solving quadratic equations is to solve the simpler equations we obtained by setting factors to zero. From our equations

$x-10=0$
$x-8=0$

, we solve for $x$ to get:

$x = 10$
$x = 8$

. These are the solutions to the original quadratic equation $x^2 - 18x + 80 = 0$. Therefore, $x = 10$ and $x = 8$ are the values that satisfy our equation. This completes our process of solving a quadratic equation by factoring.

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91影视

Solve each equation. $$ x^{2}-18 x+80=0 $$

Short Answer

Step by step solution

Identify the equation type

Factor the quadratic equation

Set each factor to zero

Solve for \(x\)

Key Concepts

Factoring Quadratic Equations

Solving Quadratic Equations

Setting Factors to Zero

Solutions to Quadratic Equations

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