Solve (x^2-12x^2+23x-5)+(x-3)

$$( x ^ { 2 } - 12 x ^ { 2 } + 23 x - 5 ) + ( x - 3 )$$

Evaluate

$$-11x^{2}+24x-8$$

Solution Steps

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}+23x-5+x-3$$

Combine 23x and x to get 24x.

$$-11x^{2}+24x-5-3$$

Subtract 3 from -5 to get -8.

$$-11x^{2}+24x-8$$

Factor

$$-11\left(x-\frac{12-2\sqrt{14}}{11}\right)\left(x-\frac{2\sqrt{14}+12}{11}\right)$$

Steps Using the Quadratic Formula

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$factor(-11x^{2}+23x-5+x-3)$$

Combine 23x and x to get 24x.

$$factor(-11x^{2}+24x-5-3)$$

Subtract 3 from -5 to get -8.

$$factor(-11x^{2}+24x-8)$$

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.

$$-11x^{2}+24x-8=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-11\right)\left(-8\right)}}{2\left(-11\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-11\right)\left(-8\right)}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{-24±\sqrt{576+44\left(-8\right)}}{2\left(-11\right)}$$

Multiply 44 times -8.

$$x=\frac{-24±\sqrt{576-352}}{2\left(-11\right)}$$

Add 576 to -352.

$$x=\frac{-24±\sqrt{224}}{2\left(-11\right)}$$

Take the square root of 224.

$$x=\frac{-24±4\sqrt{14}}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{-24±4\sqrt{14}}{-22}$$

Now solve the equation x=\frac{-24±4\sqrt{14}}{-22} when ± is plus. Add -24 to 4\sqrt{14}.

$$x=\frac{4\sqrt{14}-24}{-22}$$

Divide -24+4\sqrt{14} by -22.

$$x=\frac{12-2\sqrt{14}}{11}$$

Now solve the equation x=\frac{-24±4\sqrt{14}}{-22} when ± is minus. Subtract 4\sqrt{14} from -24.

$$x=\frac{-4\sqrt{14}-24}{-22}$$

Divide -24-4\sqrt{14} by -22.

$$x=\frac{2\sqrt{14}+12}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{12-2\sqrt{14}}{11} for x_{1} and \frac{12+2\sqrt{14}}{11} for x_{2}.

$$-11x^{2}+24x-8=-11\left(x-\frac{12-2\sqrt{14}}{11}\right)\left(x-\frac{2\sqrt{14}+12}{11}\right)$$