Solve f^2-8f-9

$$f ^ { 2 } - 8 f - 9$$

Factor

$$\left(f-9\right)\left(f+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as f^{2}+af+bf-9. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-9\right)=-9$$

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 -9.

$$1,-9 3,-3$$

Calculate the sum for each pair.

$$1-9=-8 3-3=0$$

The solution is the pair that gives sum -8.

$$a=-9 b=1$$

Rewrite f^{2}-8f-9 as \left(f^{2}-9f\right)+\left(f-9\right).

$$\left(f^{2}-9f\right)+\left(f-9\right)$$

Factor out f in f^{2}-9f.

$$f\left(f-9\right)+f-9$$

Factor out common term f-9 by using distributive property.

$$\left(f-9\right)\left(f+1\right)$$

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.

$$f^{2}-8f-9=0$$

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.

$$f=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-9\right)}}{2}$$

Square -8.

$$f=\frac{-\left(-8\right)±\sqrt{64-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$f=\frac{-\left(-8\right)±\sqrt{64+36}}{2}$$

Add 64 to 36.

$$f=\frac{-\left(-8\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$f=\frac{-\left(-8\right)±10}{2}$$

The opposite of -8 is 8.

$$f=\frac{8±10}{2}$$

Now solve the equation f=\frac{8±10}{2} when ± is plus. Add 8 to 10.

$$f=\frac{18}{2}$$

Divide 18 by 2.

$$f=9$$

Now solve the equation f=\frac{8±10}{2} when ± is minus. Subtract 10 from 8.

$$f=-\frac{2}{2}$$

Divide -2 by 2.

$$f=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -1 for x_{2}.

$$f^{2}-8f-9=\left(f-9\right)\left(f-\left(-1\right)\right)$$

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

$$f^{2}-8f-9=\left(f-9\right)\left(f+1\right)$$

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.

$$x ^ 2 -8x -9 = 0$$

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

$$r + s = 8 rs = -9$$

Two numbers r and s sum up to 8 exactly when the average of the two numbers is \frac{1}{2}*8 = 4. 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.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 4 - u s = 4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -9

$$(4 - u) (4 + u) = -9$$

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

$$16 - u^2 = -9$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = -9-16 = -25$$

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.

$$r =4 - 5 = -1 s = 4 + 5 = 9$$

Evaluate

$$\left(f-9\right)\left(f+1\right)$$