Solve 19(4x-7)(2x+3)-8x(1-4)=0

$$19 ( 4 x - 7 ) ( 2 x + 3 ) - 8 x ( 1 - 4 ) = 0$$

Solve for x

$$x = \frac{\sqrt{60697} + 7}{152} \approx 1.666892181$$

$$x=\frac{7-\sqrt{60697}}{152}\approx -1.574786918$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 19 by 4x-7.

$$\left(76x-133\right)\left(2x+3\right)-8x\left(1-4\right)=0$$

Use the distributive property to multiply 76x-133 by 2x+3 and combine like terms.

$$152x^{2}-38x-399-8x\left(1-4\right)=0$$

Subtract 4 from 1 to get -3.

$$152x^{2}-38x-399-8x\left(-3\right)=0$$

Multiply 8 and -3 to get -24.

$$152x^{2}-38x-399-\left(-24x\right)=0$$

The opposite of -24x is 24x.

$$152x^{2}-38x-399+24x=0$$

Combine -38x and 24x to get -14x.

$$152x^{2}-14x-399=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 152 for a, -14 for b, and -399 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 152\left(-399\right)}}{2\times 152}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 152\left(-399\right)}}{2\times 152}$$

Multiply -4 times 152.

$$x=\frac{-\left(-14\right)±\sqrt{196-608\left(-399\right)}}{2\times 152}$$

Multiply -608 times -399.

$$x=\frac{-\left(-14\right)±\sqrt{196+242592}}{2\times 152}$$

Add 196 to 242592.

$$x=\frac{-\left(-14\right)±\sqrt{242788}}{2\times 152}$$

Take the square root of 242788.

$$x=\frac{-\left(-14\right)±2\sqrt{60697}}{2\times 152}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{60697}}{2\times 152}$$

Multiply 2 times 152.

$$x=\frac{14±2\sqrt{60697}}{304}$$

Now solve the equation x=\frac{14±2\sqrt{60697}}{304} when ± is plus. Add 14 to 2\sqrt{60697}.

$$x=\frac{2\sqrt{60697}+14}{304}$$

Divide 14+2\sqrt{60697} by 304.

$$x=\frac{\sqrt{60697}+7}{152}$$

Now solve the equation x=\frac{14±2\sqrt{60697}}{304} when ± is minus. Subtract 2\sqrt{60697} from 14.

$$x=\frac{14-2\sqrt{60697}}{304}$$

Divide 14-2\sqrt{60697} by 304.

$$x=\frac{7-\sqrt{60697}}{152}$$

The equation is now solved.

$$x=\frac{\sqrt{60697}+7}{152} x=\frac{7-\sqrt{60697}}{152}$$

Steps for Completing the Square

Use the distributive property to multiply 19 by 4x-7.

$$\left(76x-133\right)\left(2x+3\right)-8x\left(1-4\right)=0$$

Use the distributive property to multiply 76x-133 by 2x+3 and combine like terms.

$$152x^{2}-38x-399-8x\left(1-4\right)=0$$

Subtract 4 from 1 to get -3.

$$152x^{2}-38x-399-8x\left(-3\right)=0$$

Multiply 8 and -3 to get -24.

$$152x^{2}-38x-399-\left(-24x\right)=0$$

The opposite of -24x is 24x.

$$152x^{2}-38x-399+24x=0$$

Combine -38x and 24x to get -14x.

$$152x^{2}-14x-399=0$$

Add 399 to both sides. Anything plus zero gives itself.

$$152x^{2}-14x=399$$

Divide both sides by 152.

$$\frac{152x^{2}-14x}{152}=\frac{399}{152}$$

Dividing by 152 undoes the multiplication by 152.

$$x^{2}+\left(-\frac{14}{152}\right)x=\frac{399}{152}$$

Reduce the fraction \frac{-14}{152} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{76}x=\frac{399}{152}$$

Reduce the fraction \frac{399}{152} to lowest terms by extracting and canceling out 19.

$$x^{2}-\frac{7}{76}x=\frac{21}{8}$$

Divide -\frac{7}{76}, the coefficient of the x term, by 2 to get -\frac{7}{152}. Then add the square of -\frac{7}{152} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{76}x+\left(-\frac{7}{152}\right)^{2}=\frac{21}{8}+\left(-\frac{7}{152}\right)^{2}$$

Square -\frac{7}{152} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{76}x+\frac{49}{23104}=\frac{21}{8}+\frac{49}{23104}$$

Add \frac{21}{8} to \frac{49}{23104} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{76}x+\frac{49}{23104}=\frac{60697}{23104}$$

Factor x^{2}-\frac{7}{76}x+\frac{49}{23104}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{152}\right)^{2}=\frac{60697}{23104}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{152}\right)^{2}}=\sqrt{\frac{60697}{23104}}$$

Simplify.

$$x-\frac{7}{152}=\frac{\sqrt{60697}}{152} x-\frac{7}{152}=-\frac{\sqrt{60697}}{152}$$

Add \frac{7}{152} to both sides of the equation.

$$x=\frac{\sqrt{60697}+7}{152} x=\frac{7-\sqrt{60697}}{152}$$