Solve 3x^2-7=-4x

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

Solve for x

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$x=1$$

Steps Using Factoring By Grouping

Add 4x to both sides.

$$3x^{2}-7+4x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$3x^{2}+4x-7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=4 ab=3\left(-7\right)=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 4.

$$a=-3 b=7$$

Rewrite 3x^{2}+4x-7 as \left(3x^{2}-3x\right)+\left(7x-7\right).

$$\left(3x^{2}-3x\right)+\left(7x-7\right)$$

Factor out 3x in the first and 7 in the second group.

$$3x\left(x-1\right)+7\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(3x+7\right)$$

To find equation solutions, solve x-1=0 and 3x+7=0.

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

Steps Using the Quadratic Formula

Add 4x to both sides.

$$3x^{2}-7+4x=0$$

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.

$$3x^{2}+4x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 3\left(-7\right)}}{2\times 3}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 3\left(-7\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-4±\sqrt{16-12\left(-7\right)}}{2\times 3}$$

Multiply -12 times -7.

$$x=\frac{-4±\sqrt{16+84}}{2\times 3}$$

Add 16 to 84.

$$x=\frac{-4±\sqrt{100}}{2\times 3}$$

Take the square root of 100.

$$x=\frac{-4±10}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-4±10}{6}$$

Now solve the equation x=\frac{-4±10}{6} when ± is plus. Add -4 to 10.

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

Divide 6 by 6.

$$x=1$$

Now solve the equation x=\frac{-4±10}{6} when ± is minus. Subtract 10 from -4.

$$x=-\frac{14}{6}$$

Reduce the fraction \frac{-14}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{3}$$

The equation is now solved.

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

Steps for Completing the Square

Add 4x to both sides.

$$3x^{2}-7+4x=0$$

Add 7 to both sides. Anything plus zero gives itself.

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

Divide both sides by 3.

$$\frac{3x^{2}+4x}{3}=\frac{7}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{4}{3}x=\frac{7}{3}$$

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{7}{3}+\frac{4}{9}$$

Add \frac{7}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{25}{9}$$

Factor x^{2}+\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{3}\right)^{2}=\frac{25}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{\frac{25}{9}}$$

Simplify.

$$x+\frac{2}{3}=\frac{5}{3} x+\frac{2}{3}=-\frac{5}{3}$$

Subtract \frac{2}{3} from both sides of the equation.

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