Solve 7x(x-5)=10-2x

$$7 x ( x - 5 ) = 10 - 2 x$$

Solve for x

$$x=-\frac{2}{7}\approx -0.285714286$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7x by x-5.

$$7x^{2}-35x=10-2x$$

Subtract 10 from both sides.

$$7x^{2}-35x-10=-2x$$

Add 2x to both sides.

$$7x^{2}-35x-10+2x=0$$

Combine -35x and 2x to get -33x.

$$7x^{2}-33x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -33 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 7\left(-10\right)}}{2\times 7}$$

Square -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\times 7\left(-10\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-33\right)±\sqrt{1089-28\left(-10\right)}}{2\times 7}$$

Multiply -28 times -10.

$$x=\frac{-\left(-33\right)±\sqrt{1089+280}}{2\times 7}$$

Add 1089 to 280.

$$x=\frac{-\left(-33\right)±\sqrt{1369}}{2\times 7}$$

Take the square root of 1369.

$$x=\frac{-\left(-33\right)±37}{2\times 7}$$

The opposite of -33 is 33.

$$x=\frac{33±37}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{33±37}{14}$$

Now solve the equation x=\frac{33±37}{14} when ± is plus. Add 33 to 37.

$$x=\frac{70}{14}$$

Divide 70 by 14.

$$x=5$$

Now solve the equation x=\frac{33±37}{14} when ± is minus. Subtract 37 from 33.

$$x=-\frac{4}{14}$$

Reduce the fraction \frac{-4}{14} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 7x by x-5.

$$7x^{2}-35x=10-2x$$

Add 2x to both sides.

$$7x^{2}-35x+2x=10$$

Combine -35x and 2x to get -33x.

$$7x^{2}-33x=10$$

Divide both sides by 7.

$$\frac{7x^{2}-33x}{7}=\frac{10}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{33}{7}x=\frac{10}{7}$$

Divide -\frac{33}{7}, the coefficient of the x term, by 2 to get -\frac{33}{14}. Then add the square of -\frac{33}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{33}{7}x+\left(-\frac{33}{14}\right)^{2}=\frac{10}{7}+\left(-\frac{33}{14}\right)^{2}$$

Square -\frac{33}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{33}{7}x+\frac{1089}{196}=\frac{10}{7}+\frac{1089}{196}$$

Add \frac{10}{7} to \frac{1089}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{33}{7}x+\frac{1089}{196}=\frac{1369}{196}$$

Factor x^{2}-\frac{33}{7}x+\frac{1089}{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}.

$$\left(x-\frac{33}{14}\right)^{2}=\frac{1369}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{33}{14}\right)^{2}}=\sqrt{\frac{1369}{196}}$$

Simplify.

$$x-\frac{33}{14}=\frac{37}{14} x-\frac{33}{14}=-\frac{37}{14}$$

Add \frac{33}{14} to both sides of the equation.

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