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Chapter 6: Problem 78

Solve each equation by the method of your choice. \(7 x(x-2)=3-2(x+4)\)

Short Answer

Expert verified
The solutions to the equation are \(x_1 = 1\) and \(x_2 = \frac{5}{7}\).

Step by step solution

01

Simplify Both Sides

Start by expanding and simplifying both sides of the equation. The left side becomes \(7x^2 - 14x\), and the right side simplifies to \(3 - 2x - 8\) which further simplifies to \(-2x - 5\). Now our equation is \(7x^2 - 14x = -2x - 5\).
02

Rearrange the Equation

Rearrange the equation to set it equal to zero. This involves adding \(2x\) and \(5\) to both sides to isolate the variable terms on one side and constant on the other side. This gives us \(7x^2 - 14x + 2x + 5 = 0\) which further simplifies to \(7x^2 - 12x + 5 = 0\).
03

Solve Quadratic

Now we have a standard form quadratic equation, and we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), to solve for 'x' where 'a', 'b' and 'c' are the coefficients from the quadratic equation. In this case, 'a' is 7, 'b' is -12, and 'c' is 5. Plugging these in gives us two values for 'x' which are \(x_1 = 1\) and \(x_2 = \frac{5}{7}\).

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Most popular questions from this chapter

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

Solve the equations using the quadratic formula. \(x^{2}+2 x=4\)

Solve the quadratic equations by factoring. \(x^{2}-11 x=-10\)

Factor the trinomials , or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(3 x^{2}-2 x-5\)

Solve the equations using the quadratic formula. \(3 x^{2}=5 x-1\)
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