Solve 8*1*3left(2x-5right)^2-147=0

$$8 \cdot 1 \cdot 3 { \left(2x-5 \right) }^{ 2 } -147=0$$

Solve for x

$$x=\frac{7\sqrt{2}}{8}+\frac{5}{2}\approx 3.737436867$$

$$x=-\frac{7\sqrt{2}}{8}+\frac{5}{2}\approx 1.262563133$$

Steps Using the Quadratic Formula

Multiply 8 and 1 to get 8.

$$8\times 3\left(2x-5\right)^{2}-147=0$$

Multiply 8 and 3 to get 24.

$$24\left(2x-5\right)^{2}-147=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$24\left(4x^{2}-20x+25\right)-147=0$$

Use the distributive property to multiply 24 by 4x^{2}-20x+25.

$$96x^{2}-480x+600-147=0$$

Subtract 147 from 600 to get 453.

$$96x^{2}-480x+453=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 96 for a, -480 for b, and 453 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-480\right)±\sqrt{\left(-480\right)^{2}-4\times 96\times 453}}{2\times 96}$$

Square -480.

$$x=\frac{-\left(-480\right)±\sqrt{230400-4\times 96\times 453}}{2\times 96}$$

Multiply -4 times 96.

$$x=\frac{-\left(-480\right)±\sqrt{230400-384\times 453}}{2\times 96}$$

Multiply -384 times 453.

$$x=\frac{-\left(-480\right)±\sqrt{230400-173952}}{2\times 96}$$

Add 230400 to -173952.

$$x=\frac{-\left(-480\right)±\sqrt{56448}}{2\times 96}$$

Take the square root of 56448.

$$x=\frac{-\left(-480\right)±168\sqrt{2}}{2\times 96}$$

The opposite of -480 is 480.

$$x=\frac{480±168\sqrt{2}}{2\times 96}$$

Multiply 2 times 96.

$$x=\frac{480±168\sqrt{2}}{192}$$

Now solve the equation x=\frac{480±168\sqrt{2}}{192} when ± is plus. Add 480 to 168\sqrt{2}.

$$x=\frac{168\sqrt{2}+480}{192}$$

Divide 480+168\sqrt{2} by 192.

$$x=\frac{7\sqrt{2}}{8}+\frac{5}{2}$$

Now solve the equation x=\frac{480±168\sqrt{2}}{192} when ± is minus. Subtract 168\sqrt{2} from 480.

$$x=\frac{480-168\sqrt{2}}{192}$$

Divide 480-168\sqrt{2} by 192.

$$x=-\frac{7\sqrt{2}}{8}+\frac{5}{2}$$

The equation is now solved.

$$x=\frac{7\sqrt{2}}{8}+\frac{5}{2} x=-\frac{7\sqrt{2}}{8}+\frac{5}{2}$$

Steps for Completing the Square

Multiply 8 and 1 to get 8.

$$8\times 3\left(2x-5\right)^{2}-147=0$$

Multiply 8 and 3 to get 24.

$$24\left(2x-5\right)^{2}-147=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$24\left(4x^{2}-20x+25\right)-147=0$$

Use the distributive property to multiply 24 by 4x^{2}-20x+25.

$$96x^{2}-480x+600-147=0$$

Subtract 147 from 600 to get 453.

$$96x^{2}-480x+453=0$$

Subtract 453 from both sides. Anything subtracted from zero gives its negation.

$$96x^{2}-480x=-453$$

Divide both sides by 96.

$$\frac{96x^{2}-480x}{96}=-\frac{453}{96}$$

Dividing by 96 undoes the multiplication by 96.

$$x^{2}+\left(-\frac{480}{96}\right)x=-\frac{453}{96}$$

Divide -480 by 96.

$$x^{2}-5x=-\frac{453}{96}$$

Reduce the fraction \frac{-453}{96} to lowest terms by extracting and canceling out 3.

$$x^{2}-5x=-\frac{151}{32}$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{151}{32}+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-\frac{151}{32}+\frac{25}{4}$$

Add -\frac{151}{32} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5x+\frac{25}{4}=\frac{49}{32}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{49}{32}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{49}{32}}$$

Simplify.

$$x-\frac{5}{2}=\frac{7\sqrt{2}}{8} x-\frac{5}{2}=-\frac{7\sqrt{2}}{8}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{7\sqrt{2}}{8}+\frac{5}{2} x=-\frac{7\sqrt{2}}{8}+\frac{5}{2}$$