Solve 3f^2-24f+21

$$3 f ^ { 2 } - 24 f + 21$$

Factor

$$3\left(f-7\right)\left(f-1\right)$$

Solution Steps

Factor out 3.

$$3\left(f^{2}-8f+7\right)$$

Consider f^{2}-8f+7. Factor the expression by grouping. First, the expression needs to be rewritten as f^{2}+af+bf+7. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\times 7=7$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-7 b=-1$$

Rewrite f^{2}-8f+7 as \left(f^{2}-7f\right)+\left(-f+7\right).

$$\left(f^{2}-7f\right)+\left(-f+7\right)$$

Factor out f in the first and -1 in the second group.

$$f\left(f-7\right)-\left(f-7\right)$$

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

$$\left(f-7\right)\left(f-1\right)$$

Rewrite the complete factored expression.

$$3\left(f-7\right)\left(f-1\right)$$

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.

$$3f^{2}-24f+21=0$$

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.

$$f=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 3\times 21}}{2\times 3}$$

Square -24.

$$f=\frac{-\left(-24\right)±\sqrt{576-4\times 3\times 21}}{2\times 3}$$

Multiply -4 times 3.

$$f=\frac{-\left(-24\right)±\sqrt{576-12\times 21}}{2\times 3}$$

Multiply -12 times 21.

$$f=\frac{-\left(-24\right)±\sqrt{576-252}}{2\times 3}$$

Add 576 to -252.

$$f=\frac{-\left(-24\right)±\sqrt{324}}{2\times 3}$$

Take the square root of 324.

$$f=\frac{-\left(-24\right)±18}{2\times 3}$$

The opposite of -24 is 24.

$$f=\frac{24±18}{2\times 3}$$

Multiply 2 times 3.

$$f=\frac{24±18}{6}$$

Now solve the equation f=\frac{24±18}{6} when ± is plus. Add 24 to 18.

$$f=\frac{42}{6}$$

Divide 42 by 6.

$$f=7$$

Now solve the equation f=\frac{24±18}{6} when ± is minus. Subtract 18 from 24.

$$f=\frac{6}{6}$$

Divide 6 by 6.

$$f=1$$

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

$$3f^{2}-24f+21=3\left(f-7\right)\left(f-1\right)$$

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.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 -8x +7 = 0$$

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

$$r + s = 8 rs = 7$$

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 = 7

$$(4 - u) (4 + u) = 7$$

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

$$16 - u^2 = 7$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = 7-16 = -9$$

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

$$u^2 = 9 u = \pm\sqrt{9} = \pm 3$$

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

$$r =4 - 3 = 1 s = 4 + 3 = 7$$

Evaluate

$$3\left(f-7\right)\left(f-1\right)$$