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Chapter 6: Problem 69

(a) solve. (b) check. $$ x^{2}+x=90 $$

Short Answer

Expert verified
x = 9 and x = -10.

Step by step solution

01

- Move Equation to Standard Form

First, let's move all terms to one side of the equation to set it to zero. The given equation is: x^2 + x = 90 Subtract 90 from both sides: x^2 + x - 90 = 0
02

- Factor the Quadratic Equation

Next, factor the quadratic equation. We look for two numbers that multiply to -90 and add up to 1 (the coefficient of x). These numbers are -9 and 10. So, the factored form is: (x - 9)(x + 10) = 0
03

- Solve for x

Set each factor to zero to solve for x: x - 9 = 0 and x + 10 = 0 Solving these equations: x = 9 x = -10
04

- Check the Solutions

Substitute x = 9 and x = -10 back into the original equation to ensure they satisfy it. For x = 9: (9)^2 + 9 = 81 + 9 = 90 (True) For x = -10: (-10)^2 + (-10) = 100 - 10 = 90 (True)

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Key Concepts

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

Factoring Quadratics
Factoring quadratics is a method used to solve quadratic equations. A quadratic equation typically looks like this: \[ ax^2 + bx + c = 0 \] Here, 'a', 'b', and 'c' are coefficients. To factor, we seek two numbers that multiply to 'a*c' and add up to 'b'. These numbers split the middle term. In our example: \[ x^2 + x - 90 = 0 \] we look for factors of -90 that add up to 1. The numbers are -9 and 10. Thus, we can write: \[ x^2 + 10x - 9x - 90 \] Grouping terms, we get: \[ (x + 10)(x - 9) = 0 \] and we've factored the equation!
This factoring allows us to solve the equation more easily.
Standard Form of Quadratic Equation
Quadratic equations are often presented in standard form. This form is: \[ ax^2 + bx + c = 0 \] where 'a', 'b', and 'c' are constants. Standard form helps us use different methods like factoring, completing the square, and the quadratic formula to solve these equations. For instance, take the equation \[ x^2 + x = 90 \]. To convert it to standard form, we move all terms to one side to set the equation to zero. Subtracting 90 from both sides results in: \[ x^2 + x - 90 = 0 \] It's now in standard form, ready for solving.
Checking Solutions in Algebra
After solving a quadratic equation, it's essential to verify the solutions. We do this by substituting the values back into the original equation to see if they satisfy it. In our example, solving \[ x^2 + x - 90 = 0 \] gave us solutions 'x = 9' and 'x = -10'. Substituting 'x = 9' back: \( 9^2 + 9 = 81 + 9 = 90 \) is true, so it's a solution. Substituting 'x = -10' back: \( (-10)^2 + (-10) = 100 - 10 = 90 \) also holds true.
Therefore, both solutions are correct. Always check your solutions to ensure they are valid.

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Most popular questions from this chapter

Solve. $$ (h+2)(h-10)=0 $$

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