Solve x^2+3x-4x^2+7x+12

$$x ^ { 2 } + 3 x - 4 x ^ { 2 } + 7 x + 12$$

Evaluate

$$12+10x-3x^{2}$$

Solution Steps

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

$$-3x^{2}+3x+7x+12$$

Combine 3x and 7x to get 10x.

$$-3x^{2}+10x+12$$

Factor

$$-3\left(x-\frac{5-\sqrt{61}}{3}\right)\left(x-\frac{\sqrt{61}+5}{3}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-3x^{2}+3x+7x+12)$$

Combine 3x and 7x to get 10x.

$$factor(-3x^{2}+10x+12)$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-3x^{2}+10x+12=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-10±\sqrt{100+12\times 12}}{2\left(-3\right)}$$

Multiply 12 times 12.

$$x=\frac{-10±\sqrt{100+144}}{2\left(-3\right)}$$

Add 100 to 144.

$$x=\frac{-10±\sqrt{244}}{2\left(-3\right)}$$

Take the square root of 244.

$$x=\frac{-10±2\sqrt{61}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-10±2\sqrt{61}}{-6}$$

Now solve the equation x=\frac{-10±2\sqrt{61}}{-6} when ± is plus. Add -10 to 2\sqrt{61}.

$$x=\frac{2\sqrt{61}-10}{-6}$$

Divide -10+2\sqrt{61} by -6.

$$x=\frac{5-\sqrt{61}}{3}$$

Now solve the equation x=\frac{-10±2\sqrt{61}}{-6} when ± is minus. Subtract 2\sqrt{61} from -10.

$$x=\frac{-2\sqrt{61}-10}{-6}$$

Divide -10-2\sqrt{61} by -6.

$$x=\frac{\sqrt{61}+5}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{61}}{3} for x_{1} and \frac{5+\sqrt{61}}{3} for x_{2}.

$$-3x^{2}+10x+12=-3\left(x-\frac{5-\sqrt{61}}{3}\right)\left(x-\frac{\sqrt{61}+5}{3}\right)$$