Solve 3x-x^2-3x-10/1+(x-2/4)^2=0

$$3 x - \frac { x ^ { 2 } - 3 x - 10 } { 1 } + ( \frac { x - 2 } { 4 } ) ^ { 2 } = 0$$

Solve for x

$$x = \frac{4 \sqrt{286} + 46}{15} \approx 7.576409207$$

$$x=\frac{46-4\sqrt{286}}{15}\approx -1.443075873$$

Steps Using the Quadratic Formula

Anything divided by one gives itself.

$$3x-\left(x^{2}-3x-10\right)+\left(\frac{x-2}{4}\right)^{2}=0$$

To find the opposite of x^{2}-3x-10, find the opposite of each term.

$$3x-x^{2}+3x+10+\left(\frac{x-2}{4}\right)^{2}=0$$

Combine 3x and 3x to get 6x.

$$6x-x^{2}+10+\left(\frac{x-2}{4}\right)^{2}=0$$

To raise \frac{x-2}{4} to a power, raise both numerator and denominator to the power and then divide.

$$6x-x^{2}+10+\frac{\left(x-2\right)^{2}}{4^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6x-x^{2}+10 times \frac{4^{2}}{4^{2}}.

$$\frac{\left(6x-x^{2}+10\right)\times 4^{2}}{4^{2}}+\frac{\left(x-2\right)^{2}}{4^{2}}=0$$

Since \frac{\left(6x-x^{2}+10\right)\times 4^{2}}{4^{2}} and \frac{\left(x-2\right)^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(6x-x^{2}+10\right)\times 4^{2}+\left(x-2\right)^{2}}{4^{2}}=0$$

Do the multiplications in \left(6x-x^{2}+10\right)\times 4^{2}+\left(x-2\right)^{2}.

$$\frac{96x-16x^{2}+160+x^{2}-4x+4}{4^{2}}=0$$

Combine like terms in 96x-16x^{2}+160+x^{2}-4x+4.

$$\frac{92x-15x^{2}+164}{4^{2}}=0$$

Calculate 4 to the power of 2 and get 16.

$$\frac{92x-15x^{2}+164}{16}=0$$

Divide each term of 92x-15x^{2}+164 by 16 to get \frac{23}{4}x-\frac{15}{16}x^{2}+\frac{41}{4}.

$$\frac{23}{4}x-\frac{15}{16}x^{2}+\frac{41}{4}=0$$

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.

$$-\frac{15}{16}x^{2}+\frac{23}{4}x+\frac{41}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{15}{16} for a, \frac{23}{4} for b, and \frac{41}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{23}{4}±\sqrt{\left(\frac{23}{4}\right)^{2}-4\left(-\frac{15}{16}\right)\times \frac{41}{4}}}{2\left(-\frac{15}{16}\right)}$$

Square \frac{23}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{23}{4}±\sqrt{\frac{529}{16}-4\left(-\frac{15}{16}\right)\times \frac{41}{4}}}{2\left(-\frac{15}{16}\right)}$$

Multiply -4 times -\frac{15}{16}.

$$x=\frac{-\frac{23}{4}±\sqrt{\frac{529}{16}+\frac{15}{4}\times \frac{41}{4}}}{2\left(-\frac{15}{16}\right)}$$

Multiply \frac{15}{4} times \frac{41}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{23}{4}±\sqrt{\frac{529+615}{16}}}{2\left(-\frac{15}{16}\right)}$$

Add \frac{529}{16} to \frac{615}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{23}{4}±\sqrt{\frac{143}{2}}}{2\left(-\frac{15}{16}\right)}$$

Take the square root of \frac{143}{2}.

$$x=\frac{-\frac{23}{4}±\frac{\sqrt{286}}{2}}{2\left(-\frac{15}{16}\right)}$$

Multiply 2 times -\frac{15}{16}.

$$x=\frac{-\frac{23}{4}±\frac{\sqrt{286}}{2}}{-\frac{15}{8}}$$

Now solve the equation x=\frac{-\frac{23}{4}±\frac{\sqrt{286}}{2}}{-\frac{15}{8}} when ± is plus. Add -\frac{23}{4} to \frac{\sqrt{286}}{2}.

$$x=\frac{\frac{\sqrt{286}}{2}-\frac{23}{4}}{-\frac{15}{8}}$$

Divide -\frac{23}{4}+\frac{\sqrt{286}}{2} by -\frac{15}{8} by multiplying -\frac{23}{4}+\frac{\sqrt{286}}{2} by the reciprocal of -\frac{15}{8}.

$$x=\frac{46-4\sqrt{286}}{15}$$

Now solve the equation x=\frac{-\frac{23}{4}±\frac{\sqrt{286}}{2}}{-\frac{15}{8}} when ± is minus. Subtract \frac{\sqrt{286}}{2} from -\frac{23}{4}.

$$x=\frac{-\frac{\sqrt{286}}{2}-\frac{23}{4}}{-\frac{15}{8}}$$

Divide -\frac{23}{4}-\frac{\sqrt{286}}{2} by -\frac{15}{8} by multiplying -\frac{23}{4}-\frac{\sqrt{286}}{2} by the reciprocal of -\frac{15}{8}.

$$x=\frac{4\sqrt{286}+46}{15}$$

The equation is now solved.

$$x=\frac{46-4\sqrt{286}}{15} x=\frac{4\sqrt{286}+46}{15}$$

Steps for Completing the Square

Anything divided by one gives itself.

$$3x-\left(x^{2}-3x-10\right)+\left(\frac{x-2}{4}\right)^{2}=0$$

To find the opposite of x^{2}-3x-10, find the opposite of each term.

$$3x-x^{2}+3x+10+\left(\frac{x-2}{4}\right)^{2}=0$$

Combine 3x and 3x to get 6x.

$$6x-x^{2}+10+\left(\frac{x-2}{4}\right)^{2}=0$$

To raise \frac{x-2}{4} to a power, raise both numerator and denominator to the power and then divide.

$$6x-x^{2}+10+\frac{\left(x-2\right)^{2}}{4^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6x-x^{2}+10 times \frac{4^{2}}{4^{2}}.

$$\frac{\left(6x-x^{2}+10\right)\times 4^{2}}{4^{2}}+\frac{\left(x-2\right)^{2}}{4^{2}}=0$$

Since \frac{\left(6x-x^{2}+10\right)\times 4^{2}}{4^{2}} and \frac{\left(x-2\right)^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(6x-x^{2}+10\right)\times 4^{2}+\left(x-2\right)^{2}}{4^{2}}=0$$

Do the multiplications in \left(6x-x^{2}+10\right)\times 4^{2}+\left(x-2\right)^{2}.

$$\frac{96x-16x^{2}+160+x^{2}-4x+4}{4^{2}}=0$$

Combine like terms in 96x-16x^{2}+160+x^{2}-4x+4.

$$\frac{92x-15x^{2}+164}{4^{2}}=0$$

Calculate 4 to the power of 2 and get 16.

$$\frac{92x-15x^{2}+164}{16}=0$$

Divide each term of 92x-15x^{2}+164 by 16 to get \frac{23}{4}x-\frac{15}{16}x^{2}+\frac{41}{4}.

$$\frac{23}{4}x-\frac{15}{16}x^{2}+\frac{41}{4}=0$$

Subtract \frac{41}{4} from both sides. Anything subtracted from zero gives its negation.

$$\frac{23}{4}x-\frac{15}{16}x^{2}=-\frac{41}{4}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-\frac{15}{16}x^{2}+\frac{23}{4}x=-\frac{41}{4}$$

Divide both sides of the equation by -\frac{15}{16}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{15}{16}x^{2}+\frac{23}{4}x}{-\frac{15}{16}}=-\frac{\frac{41}{4}}{-\frac{15}{16}}$$

Dividing by -\frac{15}{16} undoes the multiplication by -\frac{15}{16}.

$$x^{2}+\frac{\frac{23}{4}}{-\frac{15}{16}}x=-\frac{\frac{41}{4}}{-\frac{15}{16}}$$

Divide \frac{23}{4} by -\frac{15}{16} by multiplying \frac{23}{4} by the reciprocal of -\frac{15}{16}.

$$x^{2}-\frac{92}{15}x=-\frac{\frac{41}{4}}{-\frac{15}{16}}$$

Divide -\frac{41}{4} by -\frac{15}{16} by multiplying -\frac{41}{4} by the reciprocal of -\frac{15}{16}.

$$x^{2}-\frac{92}{15}x=\frac{164}{15}$$

Divide -\frac{92}{15}, the coefficient of the x term, by 2 to get -\frac{46}{15}. Then add the square of -\frac{46}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{92}{15}x+\left(-\frac{46}{15}\right)^{2}=\frac{164}{15}+\left(-\frac{46}{15}\right)^{2}$$

Square -\frac{46}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{92}{15}x+\frac{2116}{225}=\frac{164}{15}+\frac{2116}{225}$$

Add \frac{164}{15} to \frac{2116}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{92}{15}x+\frac{2116}{225}=\frac{4576}{225}$$

Factor x^{2}-\frac{92}{15}x+\frac{2116}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{46}{15}\right)^{2}=\frac{4576}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{46}{15}\right)^{2}}=\sqrt{\frac{4576}{225}}$$

Simplify.

$$x-\frac{46}{15}=\frac{4\sqrt{286}}{15} x-\frac{46}{15}=-\frac{4\sqrt{286}}{15}$$

Add \frac{46}{15} to both sides of the equation.

$$x=\frac{4\sqrt{286}+46}{15} x=\frac{46-4\sqrt{286}}{15}$$