Solve 12x^2=7x+10

$$12 x ^ { 2 } = 7 x + 10$$

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using Factoring By Grouping

Subtract 7x from both sides.

$$12x^{2}-7x=10$$

Subtract 10 from both sides.

$$12x^{2}-7x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 12x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=-7 ab=12\left(-10\right)=-120$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -120.

$$1,-120 2,-60 3,-40 4,-30 5,-24 6,-20 8,-15 10,-12$$

Calculate the sum for each pair.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

The solution is the pair that gives sum -7.

$$a=-15 b=8$$

Rewrite 12x^{2}-7x-10 as \left(12x^{2}-15x\right)+\left(8x-10\right).

$$\left(12x^{2}-15x\right)+\left(8x-10\right)$$

Factor out 3x in the first and 2 in the second group.

$$3x\left(4x-5\right)+2\left(4x-5\right)$$

Factor out common term 4x-5 by using distributive property.

$$\left(4x-5\right)\left(3x+2\right)$$

To find equation solutions, solve 4x-5=0 and 3x+2=0.

$$x=\frac{5}{4} x=-\frac{2}{3}$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

$$12x^{2}-7x=10$$

Subtract 10 from both sides.

$$12x^{2}-7x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -7 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 12\left(-10\right)}}{2\times 12}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 12\left(-10\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-7\right)±\sqrt{49-48\left(-10\right)}}{2\times 12}$$

Multiply -48 times -10.

$$x=\frac{-\left(-7\right)±\sqrt{49+480}}{2\times 12}$$

Add 49 to 480.

$$x=\frac{-\left(-7\right)±\sqrt{529}}{2\times 12}$$

Take the square root of 529.

$$x=\frac{-\left(-7\right)±23}{2\times 12}$$

The opposite of -7 is 7.

$$x=\frac{7±23}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{7±23}{24}$$

Now solve the equation x=\frac{7±23}{24} when ± is plus. Add 7 to 23.

$$x=\frac{30}{24}$$

Reduce the fraction \frac{30}{24} to lowest terms by extracting and canceling out 6.

$$x=\frac{5}{4}$$

Now solve the equation x=\frac{7±23}{24} when ± is minus. Subtract 23 from 7.

$$x=-\frac{16}{24}$$

Reduce the fraction \frac{-16}{24} to lowest terms by extracting and canceling out 8.

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

The equation is now solved.

$$x=\frac{5}{4} x=-\frac{2}{3}$$

Steps for Completing the Square

Subtract 7x from both sides.

$$12x^{2}-7x=10$$

Divide both sides by 12.

$$\frac{12x^{2}-7x}{12}=\frac{10}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}-\frac{7}{12}x=\frac{10}{12}$$

Reduce the fraction \frac{10}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{12}x=\frac{5}{6}$$

Divide -\frac{7}{12}, the coefficient of the x term, by 2 to get -\frac{7}{24}. Then add the square of -\frac{7}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{12}x+\left(-\frac{7}{24}\right)^{2}=\frac{5}{6}+\left(-\frac{7}{24}\right)^{2}$$

Square -\frac{7}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{12}x+\frac{49}{576}=\frac{5}{6}+\frac{49}{576}$$

Add \frac{5}{6} to \frac{49}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{12}x+\frac{49}{576}=\frac{529}{576}$$

Factor x^{2}-\frac{7}{12}x+\frac{49}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{24}\right)^{2}=\frac{529}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{24}\right)^{2}}=\sqrt{\frac{529}{576}}$$

Simplify.

$$x-\frac{7}{24}=\frac{23}{24} x-\frac{7}{24}=-\frac{23}{24}$$

Add \frac{7}{24} to both sides of the equation.

$$x=\frac{5}{4} x=-\frac{2}{3}$$