Solve 51x-14x^2+61x-48x^2-6+23x

$$51x-14 { x }^{ 2 } +61x-48 { x }^{ 2 } -6+23x$$

Evaluate

$$-62x^{2}+135x-6$$

Solution Steps

Combine 51x and 61x to get 112x.

$$112x-14x^{2}-48x^{2}-6+23x$$

Combine -14x^{2} and -48x^{2} to get -62x^{2}.

$$112x-62x^{2}-6+23x$$

Combine 112x and 23x to get 135x.

$$135x-62x^{2}-6$$

Factor

$$-62\left(x-\frac{135-\sqrt{16737}}{124}\right)\left(x-\frac{\sqrt{16737}+135}{124}\right)$$

Steps Using the Quadratic Formula

Combine 51x and 61x to get 112x.

$$factor(112x-14x^{2}-48x^{2}-6+23x)$$

Combine -14x^{2} and -48x^{2} to get -62x^{2}.

$$factor(112x-62x^{2}-6+23x)$$

Combine 112x and 23x to get 135x.

$$factor(135x-62x^{2}-6)$$

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.

$$-62x^{2}+135x-6=0$$

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.

$$x=\frac{-135±\sqrt{135^{2}-4\left(-62\right)\left(-6\right)}}{2\left(-62\right)}$$

Square 135.

$$x=\frac{-135±\sqrt{18225-4\left(-62\right)\left(-6\right)}}{2\left(-62\right)}$$

Multiply -4 times -62.

$$x=\frac{-135±\sqrt{18225+248\left(-6\right)}}{2\left(-62\right)}$$

Multiply 248 times -6.

$$x=\frac{-135±\sqrt{18225-1488}}{2\left(-62\right)}$$

Add 18225 to -1488.

$$x=\frac{-135±\sqrt{16737}}{2\left(-62\right)}$$

Multiply 2 times -62.

$$x=\frac{-135±\sqrt{16737}}{-124}$$

Now solve the equation x=\frac{-135±\sqrt{16737}}{-124} when ± is plus. Add -135 to \sqrt{16737}.

$$x=\frac{\sqrt{16737}-135}{-124}$$

Divide -135+\sqrt{16737} by -124.

$$x=\frac{135-\sqrt{16737}}{124}$$

Now solve the equation x=\frac{-135±\sqrt{16737}}{-124} when ± is minus. Subtract \sqrt{16737} from -135.

$$x=\frac{-\sqrt{16737}-135}{-124}$$

Divide -135-\sqrt{16737} by -124.

$$x=\frac{\sqrt{16737}+135}{124}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{135-\sqrt{16737}}{124} for x_{1} and \frac{135+\sqrt{16737}}{124} for x_{2}.

$$-62x^{2}+135x-6=-62\left(x-\frac{135-\sqrt{16737}}{124}\right)\left(x-\frac{\sqrt{16737}+135}{124}\right)$$