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

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

Solve for x

$$x=5\sqrt{7}+14\approx 27.228756555$$

$$x=14-5\sqrt{7}\approx 0.771243445$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3 by x^{2}-7.

$$3x^{2}-21=4x\left(x-7\right)$$

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

$$3x^{2}-21=4x^{2}-28x$$

Subtract 4x^{2} from both sides.

$$3x^{2}-21-4x^{2}=-28x$$

Combine 3x^{2} and -4x^{2} to get -x^{2}.

$$-x^{2}-21=-28x$$

Add 28x to both sides.

$$-x^{2}-21+28x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+28x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 28 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-28±\sqrt{784+4\left(-21\right)}}{2\left(-1\right)}$$

Multiply 4 times -21.

$$x=\frac{-28±\sqrt{784-84}}{2\left(-1\right)}$$

Add 784 to -84.

$$x=\frac{-28±\sqrt{700}}{2\left(-1\right)}$$

Take the square root of 700.

$$x=\frac{-28±10\sqrt{7}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-28±10\sqrt{7}}{-2}$$

Now solve the equation x=\frac{-28±10\sqrt{7}}{-2} when ± is plus. Add -28 to 10\sqrt{7}.

$$x=\frac{10\sqrt{7}-28}{-2}$$

Divide -28+10\sqrt{7} by -2.

$$x=14-5\sqrt{7}$$

Now solve the equation x=\frac{-28±10\sqrt{7}}{-2} when ± is minus. Subtract 10\sqrt{7} from -28.

$$x=\frac{-10\sqrt{7}-28}{-2}$$

Divide -28-10\sqrt{7} by -2.

$$x=5\sqrt{7}+14$$

The equation is now solved.

$$x=14-5\sqrt{7} x=5\sqrt{7}+14$$

Steps for Completing the Square

Use the distributive property to multiply 3 by x^{2}-7.

$$3x^{2}-21=4x\left(x-7\right)$$

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

$$3x^{2}-21=4x^{2}-28x$$

Subtract 4x^{2} from both sides.

$$3x^{2}-21-4x^{2}=-28x$$

Combine 3x^{2} and -4x^{2} to get -x^{2}.

$$-x^{2}-21=-28x$$

Add 28x to both sides.

$$-x^{2}-21+28x=0$$

Add 21 to both sides. Anything plus zero gives itself.

$$-x^{2}+28x=21$$

Divide both sides by -1.

$$\frac{-x^{2}+28x}{-1}=\frac{21}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{28}{-1}x=\frac{21}{-1}$$

Divide 28 by -1.

$$x^{2}-28x=\frac{21}{-1}$$

Divide 21 by -1.

$$x^{2}-28x=-21$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-28x+\left(-14\right)^{2}=-21+\left(-14\right)^{2}$$

Square -14.

$$x^{2}-28x+196=-21+196$$

Add -21 to 196.

$$x^{2}-28x+196=175$$

Factor x^{2}-28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-14\right)^{2}=175$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{175}$$

Simplify.

$$x-14=5\sqrt{7} x-14=-5\sqrt{7}$$

Add 14 to both sides of the equation.

$$x=5\sqrt{7}+14 x=14-5\sqrt{7}$$