Solve -x^2+4x+21

$$- x ^ { 2 } + 4 x + 21$$

Factor

$$-\left(x-7\right)\left(x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=4 ab=-21=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 4.

$$a=7 b=-3$$

Rewrite -x^{2}+4x+21 as \left(-x^{2}+7x\right)+\left(-3x+21\right).

$$\left(-x^{2}+7x\right)+\left(-3x+21\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(x-7\right)-3\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x-3\right)$$

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.

$$-x^{2}+4x+21=0$$

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.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\times 21}}{2\left(-1\right)}$$

Multiply 4 times 21.

$$x=\frac{-4±\sqrt{16+84}}{2\left(-1\right)}$$

Add 16 to 84.

$$x=\frac{-4±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-4±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±10}{-2}$$

Now solve the equation x=\frac{-4±10}{-2} when ± is plus. Add -4 to 10.

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

Divide 6 by -2.

$$x=-3$$

Now solve the equation x=\frac{-4±10}{-2} when ± is minus. Subtract 10 from -4.

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

Divide -14 by -2.

$$x=7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and 7 for x_{2}.

$$-x^{2}+4x+21=-\left(x-\left(-3\right)\right)\left(x-7\right)$$

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

$$-x^{2}+4x+21=-\left(x+3\right)\left(x-7\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -4x -21 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 4 rs = -21$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -21

$$(2 - u) (2 + u) = -21$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$4 - u^2 = -21$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -21-4 = -25$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =2 - 5 = -3 s = 2 + 5 = 7$$

Evaluate

$$-\left(x-7\right)\left(x+3\right)$$