Solve 3(2x+7)^2=4(2x+7)

$$3 ( 2 x + 7 ) ^ { 2 } = 4 ( 2 x + 7 )$$

Solve for x

$$x = -\frac{17}{6} = -2\frac{5}{6} \approx -2.833333333$$

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+7\right)^{2}.

$$3\left(4x^{2}+28x+49\right)=4\left(2x+7\right)$$

Use the distributive property to multiply 3 by 4x^{2}+28x+49.

$$12x^{2}+84x+147=4\left(2x+7\right)$$

Use the distributive property to multiply 4 by 2x+7.

$$12x^{2}+84x+147=8x+28$$

Subtract 8x from both sides.

$$12x^{2}+84x+147-8x=28$$

Combine 84x and -8x to get 76x.

$$12x^{2}+76x+147=28$$

Subtract 28 from both sides.

$$12x^{2}+76x+147-28=0$$

Subtract 28 from 147 to get 119.

$$12x^{2}+76x+119=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 76 for b, and 119 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\times 12\times 119}}{2\times 12}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\times 12\times 119}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-76±\sqrt{5776-48\times 119}}{2\times 12}$$

Multiply -48 times 119.

$$x=\frac{-76±\sqrt{5776-5712}}{2\times 12}$$

Add 5776 to -5712.

$$x=\frac{-76±\sqrt{64}}{2\times 12}$$

Take the square root of 64.

$$x=\frac{-76±8}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-76±8}{24}$$

Now solve the equation x=\frac{-76±8}{24} when ± is plus. Add -76 to 8.

$$x=-\frac{68}{24}$$

Reduce the fraction \frac{-68}{24} to lowest terms by extracting and canceling out 4.

$$x=-\frac{17}{6}$$

Now solve the equation x=\frac{-76±8}{24} when ± is minus. Subtract 8 from -76.

$$x=-\frac{84}{24}$$

Reduce the fraction \frac{-84}{24} to lowest terms by extracting and canceling out 12.

$$x=-\frac{7}{2}$$

The equation is now solved.

$$x=-\frac{17}{6} x=-\frac{7}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+7\right)^{2}.

$$3\left(4x^{2}+28x+49\right)=4\left(2x+7\right)$$

Use the distributive property to multiply 3 by 4x^{2}+28x+49.

$$12x^{2}+84x+147=4\left(2x+7\right)$$

Use the distributive property to multiply 4 by 2x+7.

$$12x^{2}+84x+147=8x+28$$

Subtract 8x from both sides.

$$12x^{2}+84x+147-8x=28$$

Combine 84x and -8x to get 76x.

$$12x^{2}+76x+147=28$$

Subtract 147 from both sides.

$$12x^{2}+76x=28-147$$

Subtract 147 from 28 to get -119.

$$12x^{2}+76x=-119$$

Divide both sides by 12.

$$\frac{12x^{2}+76x}{12}=-\frac{119}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{76}{12}x=-\frac{119}{12}$$

Reduce the fraction \frac{76}{12} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{19}{3}x=-\frac{119}{12}$$

Divide \frac{19}{3}, the coefficient of the x term, by 2 to get \frac{19}{6}. Then add the square of \frac{19}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{19}{3}x+\left(\frac{19}{6}\right)^{2}=-\frac{119}{12}+\left(\frac{19}{6}\right)^{2}$$

Square \frac{19}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{19}{3}x+\frac{361}{36}=-\frac{119}{12}+\frac{361}{36}$$

Add -\frac{119}{12} to \frac{361}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{19}{3}x+\frac{361}{36}=\frac{1}{9}$$

Factor x^{2}+\frac{19}{3}x+\frac{361}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{19}{6}\right)^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{19}{6}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Simplify.

$$x+\frac{19}{6}=\frac{1}{3} x+\frac{19}{6}=-\frac{1}{3}$$

Subtract \frac{19}{6} from both sides of the equation.

$$x=-\frac{17}{6} x=-\frac{7}{2}$$