Solve (2x-1)^2=3x+11

$$( 2 x - 1 ) ^ { 2 } = 3 x + 11$$

Solve for x

$$x = \frac{\sqrt{209} + 7}{8} \approx 2.682104037$$

$$x=\frac{7-\sqrt{209}}{8}\approx -0.932104037$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1=3x+11$$

Subtract 3x from both sides.

$$4x^{2}-4x+1-3x=11$$

Combine -4x and -3x to get -7x.

$$4x^{2}-7x+1=11$$

Subtract 11 from both sides.

$$4x^{2}-7x+1-11=0$$

Subtract 11 from 1 to get -10.

$$4x^{2}-7x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -7 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 4\left(-10\right)}}{2\times 4}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 4\left(-10\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-7\right)±\sqrt{49-16\left(-10\right)}}{2\times 4}$$

Multiply -16 times -10.

$$x=\frac{-\left(-7\right)±\sqrt{49+160}}{2\times 4}$$

Add 49 to 160.

$$x=\frac{-\left(-7\right)±\sqrt{209}}{2\times 4}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{209}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{7±\sqrt{209}}{8}$$

Now solve the equation x=\frac{7±\sqrt{209}}{8} when ± is plus. Add 7 to \sqrt{209}.

$$x=\frac{\sqrt{209}+7}{8}$$

Now solve the equation x=\frac{7±\sqrt{209}}{8} when ± is minus. Subtract \sqrt{209} from 7.

$$x=\frac{7-\sqrt{209}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{209}+7}{8} x=\frac{7-\sqrt{209}}{8}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1=3x+11$$

Subtract 3x from both sides.

$$4x^{2}-4x+1-3x=11$$

Combine -4x and -3x to get -7x.

$$4x^{2}-7x+1=11$$

Subtract 1 from both sides.

$$4x^{2}-7x=11-1$$

Subtract 1 from 11 to get 10.

$$4x^{2}-7x=10$$

Divide both sides by 4.

$$\frac{4x^{2}-7x}{4}=\frac{10}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{7}{4}x=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{4}x=\frac{5}{2}$$

Divide -\frac{7}{4}, the coefficient of the x term, by 2 to get -\frac{7}{8}. Then add the square of -\frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{4}x+\left(-\frac{7}{8}\right)^{2}=\frac{5}{2}+\left(-\frac{7}{8}\right)^{2}$$

Square -\frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{5}{2}+\frac{49}{64}$$

Add \frac{5}{2} to \frac{49}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{209}{64}$$

Factor x^{2}-\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{8}\right)^{2}=\frac{209}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{\frac{209}{64}}$$

Simplify.

$$x-\frac{7}{8}=\frac{\sqrt{209}}{8} x-\frac{7}{8}=-\frac{\sqrt{209}}{8}$$

Add \frac{7}{8} to both sides of the equation.

$$x=\frac{\sqrt{209}+7}{8} x=\frac{7-\sqrt{209}}{8}$$