Given a quadratic equation, it is helpful to choose the most efficient method for solving the quadratic from our previously learned skills:

Method 1: Using square roots

We can solve quadratic equations in the form a\left(x-h\right)^{2}=k by isolating the **perfect square**, then taking the **square root** of both sides of the equation.

1 | \displaystyle a\left(x-h\right)^{2} | \displaystyle = | \displaystyle k | Given equation |

2 | \displaystyle \left(x-h\right)^{2} | \displaystyle = | \displaystyle \frac{k}{a} | Divide by a on both sides |

3 | \displaystyle x-h | \displaystyle = | \displaystyle \pm\sqrt{\frac{k}{a}} | Square root property |

4 | \displaystyle x | \displaystyle = | \displaystyle h\pm\sqrt{\frac{k}{a}} | Add h to both sides |

Following these steps, we can see that if \dfrac{k}{a} is not negative, then the equation will have real solutions. Otherwise, the equation will have no real solutions.

Method 2: Factoring

We can use the *zero product property* to solve quadratic equations by first writing the equation in factored form: a\left(x-x_{1}\right)\left(x-x_{2}\right)=0

If we can write a quadratic equation in the factored form, then we know that either x-x_{1}=0 or \\ x-x_{2}=0. This means that the solutions to the quadratic equation are x_{1} and x_{2}. This approach can be useful if the equation has rational solutions.

Method 3: Completing the square

We can use this method to rewrite a quadratic expression so that it contains a **perfect square trinomial**. A perfect square trinomial takes on the form A^{2}+2AB+B^{2}=\left(A+B\right)^{2}.

If we can rewrite an equation by completing the square, then we can solve it using square roots.

For quadratic equations where a=1, we can write them in perfect square form by following these steps:

1 | \displaystyle x^{2}+bx+c | \displaystyle = | \displaystyle 0 | Quadratic equation in standard form |

2 | \displaystyle x^{2}+bx | \displaystyle = | \displaystyle -c | Subtract c from both sides |

3 | \displaystyle x^{2}+2\left(\frac{b}{2}\right)x | \displaystyle = | \displaystyle -c | Rewrite the x coefficient |

4 | \displaystyle x^{2}+2\left(\frac{b}{2}\right)x+\left(\frac{b}{2}\right)^{2} | \displaystyle = | \displaystyle -c+\left(\frac{b}{2}\right)^{2} | Add \left(\dfrac{b}{2}\right)^{2} to both sides |

5 | \displaystyle \left(x+\frac{b}{2}\right)^{2} | \displaystyle = | \displaystyle -c+\left(\frac{b}{2}\right)^{2} | Factor the perfect square trinomial |

If a \neq 1, we can first divide through by a to factor it out.

Method 4: Graphing

If the solutions are integers, drawing the graph of the corresponding quadratic function and finding the x-intercepts is an efficient way to find the solutions.

Method 5: Using the quadratic formula

If we are unable to solve the quadratic easily using one of the previously stated methods, the **quadratic formula** is often the best approach since it can be used to solve any quadratic equation once it's written in standard form.

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

\bm{a,b,c}

the coefficients and constant from the equation f\left(x\right)=ax^{2}+bx+c

For the following quadratic equations, determine an appropriate strategy for solving, explaining your choice, and then solve for x.

a

-\dfrac{1}{2}x^{2}+x+12=0

Worked Solution

b

3\left(x-5\right)^{2}-27 = 0

Worked Solution

c

3x^{2}-5x+12=0

Worked Solution

d

9x^{2}-12x-2=0

Worked Solution

A sculpture includes a cast iron parabola, coming out of the ground, that reaches a maximum height of 2.25\text{ m}, and has a width of 6\text{ m}.

Let the position of the start of the parabola be \left(0, 0\right). Let x be the horizontal distance and y be the height of the sculpture above the ground.

Determine an appropriate quadratic function that will model the shape of the parabolic sculpture.

Worked Solution

Idea summary

Below is a list of the easiest method to use and the form of the quadratic equation for which we should use it:

Easiest equation form: | |
---|---|

Graphing | \text{Any form is fine when using technology} |

Factoring | ax^{2}+bx+c=0\text{ where }a,b,c\text{ are small} |

Square root property | x^{2}=k\text{ or }a\left(x-h\right)^{2}=k |

Completing the square | x^{2}+bx+c=0\text{ where }b\text{ is even} |

Quadratic formula | ax^{2}+bx+c=0\text{ where }a,b,c\text{ are large } |