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Chapter 6: Problem 75

Solve each equation by the method of your choice. \((x-1)(3 x+2)=-7(x-1)\)

Short Answer

Expert verified
The solutions for \(x\) are 1 and -3.

Step by step solution

01

Expand and Simplify

Start by expanding the left-hand side of the equation: \((x-1)(3x+2)\) to get \(3x^2+x-2\). And also simplifying the right-hand side: \(-7(x-1)\) to get \(-7x+7\). Now, the equation becomes \(3x^2+x-2=-7x+7\).
02

Rearrange the Equation

Move all terms to one side of the equation to set the equation to 0. It becomes \(3x^2+x+7x-2-7=0\), and simplify to get \(3x^2+8x-9=0\).
03

Solve for \(x\)

Now, we solve for \(x\). This is a quadratic equation, so we can use the quadratic formula, \(x = [-b 卤 sqrt(b^2 - 4ac)] / (2a)\), where \(a = 3\), \(b = 8\), and \(c = -9\). When you substitute \(a\), \(b\), and \(c\) into the formula, you will get \(x = [-8 卤 sqrt((8)^2 - 4*3*(-9))] / (2*3)\), or simplified further, \(x = [-8 卤 sqrt(64 + 108)] / 6 = [-8 卤 sqrt(172)] / 6\). The solutions are: \(x = 1\) and \(x = -3\).

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Most popular questions from this chapter

When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

Use the five-step strategy for solving word problems to find the number or numbers described. When a number is decreased by \(20 \%\) of itself, the result is 20 . What is the number?

In Exercises 39-40, name the property used to go from step to step each time that "(why?)" occurs. $$ \begin{aligned} &7+2(x+9) \\ &=7+(2 x+18) \quad(\text { why? }) \\ &=7+(18+2 x) \quad(\text { why? }) \\ &=(7+18)+2 x \quad(\text { why? }) \\ &=25+2 x \\ &=2 x+25 \quad \text { (why?) } \end{aligned} $$

Factor the trinomials , or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(3 x^{2}+7 x+2\)

Solve the equations using the quadratic formula. \(x^{2}-3 x=10\)
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