Solve 43x^2-12x+3x^2+2x^2+3x-6

$$43 x ^ { 2 } - 12 x + 3 x ^ { 2 } + 2 x ^ { 2 } + 3 x - 6$$

Evaluate

$$48x^{2}-9x-6$$

Solution Steps

Combine 43x^{2} and 3x^{2} to get 46x^{2}.

$$46x^{2}-12x+2x^{2}+3x-6$$

Combine 46x^{2} and 2x^{2} to get 48x^{2}.

$$48x^{2}-12x+3x-6$$

Combine -12x and 3x to get -9x.

$$48x^{2}-9x-6$$

Factor

$$48\left(x-\frac{3-\sqrt{137}}{32}\right)\left(x-\frac{\sqrt{137}+3}{32}\right)$$

Steps Using the Quadratic Formula

Combine 43x^{2} and 3x^{2} to get 46x^{2}.

$$factor(46x^{2}-12x+2x^{2}+3x-6)$$

Combine 46x^{2} and 2x^{2} to get 48x^{2}.

$$factor(48x^{2}-12x+3x-6)$$

Combine -12x and 3x to get -9x.

$$factor(48x^{2}-9x-6)$$

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.

$$48x^{2}-9x-6=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 48\left(-6\right)}}{2\times 48}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 48\left(-6\right)}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-\left(-9\right)±\sqrt{81-192\left(-6\right)}}{2\times 48}$$

Multiply -192 times -6.

$$x=\frac{-\left(-9\right)±\sqrt{81+1152}}{2\times 48}$$

Add 81 to 1152.

$$x=\frac{-\left(-9\right)±\sqrt{1233}}{2\times 48}$$

Take the square root of 1233.

$$x=\frac{-\left(-9\right)±3\sqrt{137}}{2\times 48}$$

The opposite of -9 is 9.

$$x=\frac{9±3\sqrt{137}}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{9±3\sqrt{137}}{96}$$

Now solve the equation x=\frac{9±3\sqrt{137}}{96} when ± is plus. Add 9 to 3\sqrt{137}.

$$x=\frac{3\sqrt{137}+9}{96}$$

Divide 9+3\sqrt{137} by 96.

$$x=\frac{\sqrt{137}+3}{32}$$

Now solve the equation x=\frac{9±3\sqrt{137}}{96} when ± is minus. Subtract 3\sqrt{137} from 9.

$$x=\frac{9-3\sqrt{137}}{96}$$

Divide 9-3\sqrt{137} by 96.

$$x=\frac{3-\sqrt{137}}{32}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+\sqrt{137}}{32} for x_{1} and \frac{3-\sqrt{137}}{32} for x_{2}.

$$48x^{2}-9x-6=48\left(x-\frac{\sqrt{137}+3}{32}\right)\left(x-\frac{3-\sqrt{137}}{32}\right)$$