Solving Quadratic Equations

students, quadratic equations appear everywhere in mathematics and in real life 📈🏀. A ball thrown into the air, the area of a rectangle, and many graphing problems in Functions all lead to quadratics. In this lesson, you will learn how to solve quadratic equations, how to choose the best method, and how these equations connect to graphs and functions.

By the end of this lesson, you should be able to:

explain key vocabulary such as quadratic equation, root, solution, and factor;
solve quadratic equations using factoring, the quadratic formula, completing the square, and graphing;
connect solutions of equations to the $x$-intercepts of a graph;
recognize how quadratic equations fit into the wider study of Functions.

What Is a Quadratic Equation?

A quadratic equation is an equation that can be written in the form $ax^2+bx+c=0$, where $a\neq 0$. The highest power of the variable is $2$, which is why it is called quadratic. The numbers $a$, $b$, and $c$ are constants.

The goal of solving a quadratic equation is to find the values of the variable that make the equation true. These values are called solutions or roots. In function language, if $f(x)=ax^2+bx+c$, then solving the equation $ax^2+bx+c=0$ means finding where $f(x)=0$. These are the $x$-values where the graph crosses or touches the $x$-axis.

Example:

If $x^2-5x+6=0$, we want values of $x$ that make the expression equal to zero. This equation has two solutions because a quadratic can have up to two real roots.

A useful idea is that a quadratic can be represented in several ways:

standard form: $ax^2+bx+c$
factorized form: $a(x-r_1)(x-r_2)$
vertex form: $a(x-h)^2+k$

Each form gives different information. Standard form is useful for formulas, factorized form is useful for solving, and vertex form is useful for graph transformations.

Solving by Factoring

Factoring is often the fastest method when the quadratic can be rewritten as a product of two brackets. The zero product property says that if $AB=0$, then $A=0$ or $B=0$. This is the key idea behind factoring quadratic equations.

Example 1:

Solve $x^2-5x+6=0$.

We factor the quadratic:

$$x^2-5x+6=(x-2)(x-3)$$

So the equation becomes:

$$(x-2)(x-3)=0$$

Using the zero product property:

$$x-2=0 \quad \text{or} \quad x-3=0$$

Therefore,

$$x=2 \quad \text{or} \quad x=3$$

You can check by substitution:

if $x=2$, then $2^2-5(2)+6=4-10+6=0$
if $x=3$, then $3^2-5(3)+6=9-15+6=0$

Factoring works best when numbers are friendly. If a quadratic does not factor nicely over integers, another method may be better.

A second example:

Solve $2x^2+7x+3=0$.

Factor:

$$2x^2+7x+3=(2x+1)(x+3)$$

Then set each factor equal to zero:

$$2x+1=0 \Rightarrow x=-\frac{1}{2}$$

$$x+3=0 \Rightarrow x=-3$$

So the solutions are $x=-\frac{1}{2}$ and $x=-3$.

Solving by Completing the Square

Completing the square rewrites a quadratic into vertex form. This method is useful because it connects algebra to the shape of the graph. It also leads to the quadratic formula.

Start with a quadratic in the form $x^2+bx+c=0$ or after dividing by $a$, if necessary.

Example 2:

Solve $x^2+6x+1=0$ by completing the square.

Move the constant term:

$$x^2+6x=-1$$

Take half of $6$, which is $3$, and square it to get $9$. Add $9$ to both sides:

$$x^2+6x+9=8$$

Now factor the left side:

$$(x+3)^2=8$$

Take square roots of both sides:

$$x+3=\pm\sqrt{8}$$

Since $\sqrt{8}=2\sqrt{2}$, we get:

$$x=-3\pm 2\sqrt{2}$$

So the solutions are $x=-3+2\sqrt{2}$ and $x=-3-2\sqrt{2}$.

Completing the square shows why quadratic graphs have a turning point. The expression $a(x-h)^2+k$ tells us the vertex is $(h,k)$. This is important in Functions because transformations of $y=x^2$ can shift, stretch, or reflect the graph.

Solving Using the Quadratic Formula

The quadratic formula works for every quadratic equation written as $ax^2+bx+c=0$ with $a\neq 0$.

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

This formula is powerful because it always gives the solutions, even when factoring is difficult or impossible.

Example 3:

Solve $3x^2-4x-2=0$.

Here, $a=3$, $b=-4$, and $c=-2$.

Substitute into the formula:

$$x=\frac{-(-4)\pm\sqrt{(-4)^2-4(3)(-2)}}{2(3)}$$

Simplify:

$$x=\frac{4\pm\sqrt{16+24}}{6}$$

$$x=\frac{4\pm\sqrt{40}}{6}$$

Since $\sqrt{40}=2\sqrt{10}$,

$$x=\frac{4\pm2\sqrt{10}}{6}$$

$$x=\frac{2\pm\sqrt{10}}{3}$$

So the solutions are $x=\frac{2+\sqrt{10}}{3}$ and $x=\frac{2-\sqrt{10}}{3}$.

The part under the square root, $b^2-4ac$, is called the discriminant. It tells us how many real solutions a quadratic has:

if $b^2-4ac>0$, there are two distinct real solutions;
if $b^2-4ac=0$, there is one repeated real solution;
if $b^2-4ac<0$, there are no real solutions.

This is useful in graphing. For example, if a parabola does not cross the $x$-axis, then the equation $f(x)=0$ has no real solutions.

Solving by Graphing and Interpreting Solutions

Graphs help you see solutions as intersection points with the $x$-axis. Suppose $f(x)=x^2-4x-5$. To solve $x^2-4x-5=0$, you can graph $y=x^2-4x-5$ and look for where the graph meets the $x$-axis.

For this function, the graph crosses the $x$-axis at $x=-1$ and $x=5$. So the solutions are $x=-1$ and $x=5$.

Graphing is especially helpful in IB Mathematics Analysis and Approaches SL because it connects algebra to visual reasoning. It also helps you estimate solutions when exact algebraic methods are hard to use.

A graph can show:

whether there are two, one, or no real solutions;
whether the parabola opens upward or downward;
the vertex, which is the turning point;
the axis of symmetry, which is the vertical line through the vertex.

If a quadratic is written in vertex form $y=a(x-h)^2+k$, then the vertex is $(h,k)$. The sign of $a$ tells you whether the parabola opens up or down.

Choosing a Method and Checking Your Answer

In real problem-solving, the best method depends on the equation.

Use factoring when:

the quadratic factors neatly;
you want a quick exact answer.

Use the quadratic formula when:

factoring is difficult;
the coefficients are not friendly;
you want a guaranteed method.

Use completing the square when:

you want vertex form;
you are studying graph transformations;
you need to understand the structure of the quadratic.

Always check your solutions. This is a strong habit in Functions because equations, graphs, and contexts should agree. Substitute each solution back into the original equation to make sure it works.

Real-world example 🌍:

A student throws a ball. The height of the ball after $t$ seconds might be modeled by $h(t)=-5t^2+20t+1$. To find when the ball hits the ground, solve $-5t^2+20t+1=0$. The solutions represent times when the height is zero. In context, only nonnegative values of $t$ make sense.

This shows an important point: not every algebraic solution is always meaningful in a real situation. In IB math, you must interpret answers in context.

Conclusion

Quadratic equations are a major part of Functions because they connect algebra, graphs, and real situations. students, you have seen that solving a quadratic means finding the roots of $ax^2+bx+c=0$, which are the $x$-values where the function equals zero. You can solve quadratics by factoring, completing the square, using the quadratic formula, or graphing. Each method has a purpose, and each gives insight into the structure of the function.

Understanding quadratic equations helps you read graphs, solve context problems, and move between different function representations. That is a key skill in IB Mathematics Analysis and Approaches SL ✨.

Study Notes

A quadratic equation has the form $ax^2+bx+c=0$ where $a\neq 0$.
The solutions of a quadratic are also called roots or zeros.
Solving $f(x)=0$ means finding the $x$-intercepts of the graph of $y=f(x)$.
Factoring uses the zero product property: if $AB=0$, then $A=0$ or $B=0$.
Completing the square rewrites a quadratic in the form $a(x-h)^2+k$.
The quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
The discriminant is $b^2-4ac$.
If $b^2-4ac>0$, there are two real solutions.
If $b^2-4ac=0$, there is one repeated real solution.
If $b^2-4ac<0$, there are no real solutions.
Graphs of quadratics are parabolas 📊.
The vertex form $y=a(x-h)^2+k$ shows the turning point $(h,k)$.
In context problems, always check whether the solutions make sense in the real situation.