Solve 30-7c-c^2

$$30 - 7 c - c ^ { 2 }$$

Factor

$$\left(3-c\right)\left(c+10\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-c^{2}-7c+30$$

Factor the expression by grouping. First, the expression needs to be rewritten as -c^{2}+ac+bc+30. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-30=-30$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -7.

$$a=3 b=-10$$

Rewrite -c^{2}-7c+30 as \left(-c^{2}+3c\right)+\left(-10c+30\right).

$$\left(-c^{2}+3c\right)+\left(-10c+30\right)$$

Factor out c in the first and 10 in the second group.

$$c\left(-c+3\right)+10\left(-c+3\right)$$

Factor out common term -c+3 by using distributive property.

$$\left(-c+3\right)\left(c+10\right)$$

Steps Using the Quadratic Formula

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.

$$-c^{2}-7c+30=0$$

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.

$$c=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Square -7.

$$c=\frac{-\left(-7\right)±\sqrt{49-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$c=\frac{-\left(-7\right)±\sqrt{49+4\times 30}}{2\left(-1\right)}$$

Multiply 4 times 30.

$$c=\frac{-\left(-7\right)±\sqrt{49+120}}{2\left(-1\right)}$$

Add 49 to 120.

$$c=\frac{-\left(-7\right)±\sqrt{169}}{2\left(-1\right)}$$

Take the square root of 169.

$$c=\frac{-\left(-7\right)±13}{2\left(-1\right)}$$

The opposite of -7 is 7.

$$c=\frac{7±13}{2\left(-1\right)}$$

Multiply 2 times -1.

$$c=\frac{7±13}{-2}$$

Now solve the equation c=\frac{7±13}{-2} when ± is plus. Add 7 to 13.

$$c=\frac{20}{-2}$$

Divide 20 by -2.

$$c=-10$$

Now solve the equation c=\frac{7±13}{-2} when ± is minus. Subtract 13 from 7.

$$c=-\frac{6}{-2}$$

Divide -6 by -2.

$$c=3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -10 for x_{1} and 3 for x_{2}.

$$-c^{2}-7c+30=-\left(c-\left(-10\right)\right)\left(c-3\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-c^{2}-7c+30=-\left(c+10\right)\left(c-3\right)$$

Evaluate

$$\left(3-c\right)\left(c+10\right)$$