skip to contentstylized crane logoWhite Crane Education

The Quadratic Formula, Part 1

One of the great things about quadratic equations is that there's a formula we can use to get the solutions to an equation just by plugging in the quadratic polynomial part's coefficients. Say we have a quadratic equation that looks like:

ax2 + bx + c = 0

In a real world problem, a, b and c are all going to be numbers. If we take those numbers and substitute them into this equation, we'll get the solutions that we're looking for:

$$x = \frac {-b \pm \sqrt {b^2 - 4ac}} {2a}$$

Example 1

Solve the equation x2 + 3x - 4 = 0.

First, notice that the equation is already set equal to zero. That's an important first step when using the Quadratic Formula, just like it was when we were solving by factoring.

Now we need to pick out the a, b and c values that we'll substitute into the formula. Remember that the a value is the number in front of the x2 term, the b value is the number in front of the x term and the c value is the constant at the end. This means that, for our equation, we have:

a = 1b = 3c = -4

If we substitute those values into the formula, we get:

$$x = \frac {-3 \pm \sqrt {3^2 - 4\cdot 1 \cdot (-4)}} {2 \cdot 1}$$ $$x = \frac {-3 \pm \sqrt {9 + 16}} {2}$$ $$x = \frac {-3 \pm \sqrt {25}} {2}$$ $$x = \frac {-3 \pm 5} {2}$$ $$x = \frac {-3 + 5} {2}, \frac {-3 - 5} {2}$$ $$x = \frac {2} {2}, \frac {-8} {2}$$ $$x = 1, -4$$

Example 3

Solve the equation x2 + 10x + 20 = -5.

In this example, we can't just jump right in to the formula because the equation isn't equal to 0. We have to start off by adding 5 to both sides:

x2 + 10x + 25 = 0

Now we can pull out the values of a, b and c.

a = 1b = 10c = 25

If we substitute those values into the formula, we get:

$$x = \frac {-10 \pm \sqrt {10^2 - 4\cdot 1 \cdot 25}} {2 \cdot 1}$$ $$x = \frac {-10 \pm \sqrt {100 - 100}} {2}$$ $$x = \frac {-10 \pm \sqrt {0}} {2}$$ $$x = \frac {-10} {2}$$ $$x = -5$$

Example 2

Solve the equation 2x2 - 5x - 2 = 0.

First, we need to pick out the a, b and c values that we'll substitute into the formula. Remember that the a value is the number in front of the x2 term, the b value is the number in front of the x term and the c value is the constant at the end. This means that, for our equation, we have:

a = 2b = -5c = -2

If we substitute those values into the formula, we get:

$$x = \frac {-(-5) \pm \sqrt {(-5)^2 - 4\cdot 2 \cdot (-2)}} {2 \cdot 2}$$ $$x = \frac {5 \pm \sqrt {25 + 16}} {4}$$ $$x = \frac {5 \pm \sqrt {41}} {4}$$

Since sqrt(41) can't be reduced or simplified any further. That's our final answer. It's a little messy with both a square root and a fraction left in it but these problems are often going to come out that way.

Example 4

Solve the equation x2 + 2x + 5 = 0.

Since the equation is already set equal to zero we can start by picking out the values we'll need for the formula.

a = 1b = 2c = 5

If we substitute those values into the formula, we get:

$$x = \frac {-2 \pm \sqrt {2^2 - 4\cdot 1 \cdot 5}} {2 \cdot 1}$$ $$x = \frac {-2 \pm \sqrt {4 - 20}} {2}$$ $$x = \frac {-2 \pm \sqrt {-16}} {2}$$

Now we've run into a problem. We can't take the square root of a negative number so, at this point, we have to conclude that the equation doesn't have any real number solutions. (It does have complex solutions but we'll save that for the next section.)

Dynamic Practice - Using the Quadratic Formula
 
 

link to Facebook