Solve (x-4)(17-x)=30

$$(x-4)(17-x)=30$$

Solve for x

$$x=7$$

$$x=14$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-4 by 17-x and combine like terms.

$$21x-x^{2}-68=30$$

Subtract 30 from both sides.

$$21x-x^{2}-68-30=0$$

Subtract 30 from -68 to get -98.

$$21x-x^{2}-98=0$$

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.

$$-x^{2}+21x-98=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 21 for b, and -98 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-1\right)\left(-98\right)}}{2\left(-1\right)}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\left(-1\right)\left(-98\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-21±\sqrt{441+4\left(-98\right)}}{2\left(-1\right)}$$

Multiply 4 times -98.

$$x=\frac{-21±\sqrt{441-392}}{2\left(-1\right)}$$

Add 441 to -392.

$$x=\frac{-21±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-21±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-21±7}{-2}$$

Now solve the equation x=\frac{-21±7}{-2} when ± is plus. Add -21 to 7.

$$x=-\frac{14}{-2}$$

Divide -14 by -2.

$$x=7$$

Now solve the equation x=\frac{-21±7}{-2} when ± is minus. Subtract 7 from -21.

$$x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x=14$$

The equation is now solved.

$$x=7 x=14$$

Steps for Completing the Square

Use the distributive property to multiply x-4 by 17-x and combine like terms.

$$21x-x^{2}-68=30$$

Add 68 to both sides.

$$21x-x^{2}=30+68$$

Add 30 and 68 to get 98.

$$21x-x^{2}=98$$

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.

$$-x^{2}+21x=98$$

Divide both sides by -1.

$$\frac{-x^{2}+21x}{-1}=\frac{98}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{21}{-1}x=\frac{98}{-1}$$

Divide 21 by -1.

$$x^{2}-21x=\frac{98}{-1}$$

Divide 98 by -1.

$$x^{2}-21x=-98$$

Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-98+\left(-\frac{21}{2}\right)^{2}$$

Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-21x+\frac{441}{4}=-98+\frac{441}{4}$$

Add -98 to \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{49}{4}$$

Factor x^{2}-21x+\frac{441}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{21}{2}=\frac{7}{2} x-\frac{21}{2}=-\frac{7}{2}$$

Add \frac{21}{2} to both sides of the equation.

$$x=14 x=7$$