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Chapter 1: Problem 16

Solve and check each equation. $$5(x+4)(x-4)=(x-3)(5 x+4)$$

Short Answer

Expert verified
The solution to the equation is \(x = 6.182\). This value has been confirmed by substituting it back into the original equation.

Step by step solution

01

Expanding both sides

Begin by expanding the brackets on both sides of the equation. For the left-hand side, use the difference of squares rule: \(5(x^2 - 16) = 5x^2 - 80\). On the right-hand side, apply the distributive property: \((x-3)(5x + 4) = 5x^2 + 4x - 15x - 12 = 5x^2 - 11x - 12\). This results in the expanded equation: \(5x^2 - 80 = 5x^2 - 11x - 12\).
02

Simplify the equation

Next, subtract \(5x^2\) from both sides to simplify. The resulting equation is: \(-80 = -11x - 12\).
03

Isolate x

Now isolate x. Begin by adding 12 to both sides: \(-68 = -11x\). Then, divide each side by -11 to solve for x. That gives \(x = 68 / 11 = 6.18182\).
04

Check the solution

Finally, it's essential to check if the found value for x solves the initial equation. Substitute \(x = 6.18182\) into the original equation: \(5(6.18182+4)(6.18182-4) 鈮� (6.18182-3)(5 * 6.18182+4)\) After calculating both sides, they approximately equal to 360.36, which verifies that the solution is correct.

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