Solve (x+4)*(7x-3)-x^2+16=0

$$(x+4) \times (7x-3)- { x }^{ 2 } +16=0$$

Solve for x

$$x=-4$$

$$x=-\frac{1}{6}\approx -0.166666667$$

Steps Using Factoring By Grouping

Use the distributive property to multiply x+4 by 7x-3 and combine like terms.

$$7x^{2}+25x-12-x^{2}+16=0$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}+25x-12+16=0$$

Add -12 and 16 to get 4.

$$6x^{2}+25x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=25 ab=6\times 4=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 25.

$$a=1 b=24$$

Rewrite 6x^{2}+25x+4 as \left(6x^{2}+x\right)+\left(24x+4\right).

$$\left(6x^{2}+x\right)+\left(24x+4\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(6x+1\right)+4\left(6x+1\right)$$

Factor out common term 6x+1 by using distributive property.

$$\left(6x+1\right)\left(x+4\right)$$

To find equation solutions, solve 6x+1=0 and x+4=0.

$$x=-\frac{1}{6} x=-4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+4 by 7x-3 and combine like terms.

$$7x^{2}+25x-12-x^{2}+16=0$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}+25x-12+16=0$$

Add -12 and 16 to get 4.

$$6x^{2}+25x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 25 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 6\times 4}}{2\times 6}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 6\times 4}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-25±\sqrt{625-24\times 4}}{2\times 6}$$

Multiply -24 times 4.

$$x=\frac{-25±\sqrt{625-96}}{2\times 6}$$

Add 625 to -96.

$$x=\frac{-25±\sqrt{529}}{2\times 6}$$

Take the square root of 529.

$$x=\frac{-25±23}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-25±23}{12}$$

Now solve the equation x=\frac{-25±23}{12} when ± is plus. Add -25 to 23.

$$x=-\frac{2}{12}$$

Reduce the fraction \frac{-2}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{6}$$

Now solve the equation x=\frac{-25±23}{12} when ± is minus. Subtract 23 from -25.

$$x=-\frac{48}{12}$$

Divide -48 by 12.

$$x=-4$$

The equation is now solved.

$$x=-\frac{1}{6} x=-4$$

Steps for Completing the Square

Use the distributive property to multiply x+4 by 7x-3 and combine like terms.

$$7x^{2}+25x-12-x^{2}+16=0$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}+25x-12+16=0$$

Add -12 and 16 to get 4.

$$6x^{2}+25x+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}+25x=-4$$

Divide both sides by 6.

$$\frac{6x^{2}+25x}{6}=-\frac{4}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{25}{6}x=-\frac{4}{6}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{25}{6}x=-\frac{2}{3}$$

Divide \frac{25}{6}, the coefficient of the x term, by 2 to get \frac{25}{12}. Then add the square of \frac{25}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{25}{6}x+\left(\frac{25}{12}\right)^{2}=-\frac{2}{3}+\left(\frac{25}{12}\right)^{2}$$

Square \frac{25}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{25}{6}x+\frac{625}{144}=-\frac{2}{3}+\frac{625}{144}$$

Add -\frac{2}{3} to \frac{625}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{25}{6}x+\frac{625}{144}=\frac{529}{144}$$

Factor x^{2}+\frac{25}{6}x+\frac{625}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{12}\right)^{2}=\frac{529}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Simplify.

$$x+\frac{25}{12}=\frac{23}{12} x+\frac{25}{12}=-\frac{23}{12}$$

Subtract \frac{25}{12} from both sides of the equation.

$$x=-\frac{1}{6} x=-4$$