The Quadratic Formula Explained

Algebra is arithmetic with a blank space in it. A letter such as x stands for a number you have not found yet, and solving means finding the number that makes the sentence true.

The Quadratic Formula Explained: the central idea

The quadratic formula solves every quadratic because completing the square on the general equation produces one expression for both roots.

Words you need

Quadratic equation

A quadratic equation is an equation you can write as $ax^2+bx+c=0$ with $a\ne0$, so the highest power of the variable is 2.

Standard form

Standard form is the arrangement $ax^2+bx+c=0$ with every term on one side and zero on the other, and it is the only form the quadratic formula reads correctly.

Discriminant

The discriminant is the part $b^2-4ac$ sitting under the square root, and its sign says whether there are two real roots, one repeated root, or two complex roots.

Root

A root is a value of $x$ that makes the quadratic equal zero, which is the same thing as an $x$-intercept of its graph.

Complex root

A complex root is a solution containing $i$, the number whose square is $-1$, and complex roots always arrive in a matched pair when the discriminant is negative.

Axis of symmetry

The axis of symmetry is the vertical line $x=-\dfrac{b}{2a}$ that sits exactly halfway between the two roots, which is why the formula is a midpoint plus and minus one distance.

What to know before this lesson

Know standard form $ax^2+bx+c=0$, square roots, signed substitution, and how complex numbers use $i^2=-1$.

If one of those prerequisites is uncertain, use the Algebra subject guide to locate the earlier concept before memorizing a procedure.

The quadratic formula: a worked example

Every step, with the arithmetic

1. Step 1 - Put it in standard form and read off the letters$3x^2+5x-2=0$, so $a=3$, $b=5$, $c=-2$
2. Step 2 - Compute the discriminant$D=b^2-4ac=5^2-4(3)(-2)=25+24=49$
3. Step 3 - Take the square root of the discriminant$\sqrt{49}=7$, a whole number, so both roots will be rational
4. Step 4 - Substitute into the formula$x=\dfrac{-5\pm7}{2(3)}=\dfrac{-5\pm7}{6}$
5. Step 5 - Work the plus case$x=\dfrac{-5+7}{6}=\dfrac{2}{6}=\dfrac13$
6. Step 6 - Work the minus case$x=\dfrac{-5-7}{6}=\dfrac{-12}{6}=-2$
7. Step 7 - Check one root in the original equation$3\left(\tfrac13\right)^2+5\left(\tfrac13\right)-2=\tfrac13+\tfrac53-2=0$

For $x^2-5x+6=0$, $D=25-24=1$. The two numerators are $5+1$ and $5-1$, so division by $2$ gives roots $3$ and $2$.

What the discriminant tells you before you solve

The step-by-step method for the quadratic formula

1. Rewrite the equation in standard form and identify $a$, $b$, and $c$ with their signs.
2. Compute the discriminant $D=b^2-4ac$ before taking its square root.
3. Evaluate both the plus and minus cases over the complete denominator $2a$, then simplify.

How to check your answer

Multiply $(x-2)(x-3)$ to recover $x^2-5x+6$, or substitute each root into the original equation.

Substitute a proposed value into the original statement—not only the last simplified line. For functions, also inspect domain, intercepts, and whether the graph agrees.

A mistake that changes the mathematics

The denominator is $2a$ for the entire numerator. Writing $-b\pm\sqrt D/2a$ changes the formula.

Where you will actually use this

When a thrown object lands

A ball thrown from a 5-foot ledge at 40 feet per second has height $h=-16t^2+40t+5$. Setting $h=0$ gives $D=40^2-4(-16)(5)=1600+320=1920$ and $\sqrt{1920}\approx43.8$, so $t=\dfrac{-40-43.8}{-32}\approx2.62$ seconds. The negative root is thrown away because time cannot run backwards.

Sizing a rectangular garden

With 60 meters of fencing and a target area of 200 square meters, a width $w$ gives length $30-w$, so $w(30-w)=200$ and $w^2-30w+200=0$. The discriminant is $900-800=100$, so $w=\dfrac{30\pm10}{2}$, giving 10 meters by 20 meters.

Finding a profit range

If profit is $P=-2x^2+120x-800$, setting $P=0$ gives $x^2-60x+400=0$ with discriminant $3600-1600=2000$. Since $\sqrt{2000}\approx44.7$, the break-even points are about 7.6 and 52.4 units, so the business profits between roughly 8 and 52 units.

How the quadratic formula connects to the rest of algebra

• Factoring polynomials — Factoring finds the same roots faster when the numbers are friendly, and a perfect-square discriminant is exactly the signal that the quadratic will factor over the rationals.
• Systems of equations — When a line meets a parabola, substitution turns the system into a single quadratic and the formula finishes it.
• Solving linear equations — Setting each factor equal to zero, or simplifying the final fraction, uses the same balancing moves that solve a first-degree equation.

Try a transfer problem

Apply the formula to $2x^2+3x+6=0$ and use the negative discriminant to explain why the graph has no real intercepts.

Show the worked answer

For $2x^2+3x+6=0$, $a=2$, $b=3$, $c=6$. The discriminant is $D=3^2-4(2)(6)=9-48=-39$. Because $D$ is negative, $\sqrt{D}=i\sqrt{39}$ and the roots are $x=\dfrac{-3\pm i\sqrt{39}}{4}$, or about $-0.75\pm1.56i$. No real number squares to $-39$, so no real value of $x$ makes the expression zero, and a graph of $y=2x^2+3x+6$ never touches the $x$-axis. You can see this without the formula too. Since $a=2$ is positive the parabola opens upward, and its lowest point sits at $x=-\tfrac34$ where $y=2\left(\tfrac{9}{16}\right)+3\left(-\tfrac34\right)+6=4.875$. The whole curve floats above the axis.

Once you have an answer, check it with the quadratic formula calculator. Work the problem yourself first: the calculator confirms the arithmetic, but choosing the method is the part that transfers to the next question.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to algebra practice questions.

Questions about the quadratic formula

What does a negative discriminant mean?

It means the quadratic has no real roots. The square root of a negative number is not a real number, so the two answers are complex conjugates like $\dfrac{-3+i\sqrt{39}}{4}$ and $\dfrac{-3-i\sqrt{39}}{4}$. On a graph the parabola sits entirely above or entirely below the $x$-axis and never crosses it.

Why is it $-b$ instead of $b$?

The formula comes from completing the square, which moves $b$ to the other side of the equation, so it arrives with its sign flipped. You must use the sign $b$ actually has. In $x^2-5x+6=0$ the value $b=-5$, so $-b=+5$, and the roots are $\dfrac{5\pm1}{2}$, which gives 3 and 2.

Do I have to move everything to one side first?

Yes. The formula only reads a quadratic in standard form. For $x^2+4x=12$ you must write $x^2+4x-12=0$ so that $c=-12$. Using $c=0$ from the original layout gives $x=0$ and $x=-4$, and neither one satisfies the equation.

Can I use the formula when $b$ or $c$ is missing?

Yes, a missing term just means that letter is zero. For $x^2-9=0$ take $a=1$, $b=0$, $c=-9$, so $D=0+36=36$ and $x=\dfrac{0\pm6}{2}=\pm3$. For $2x^2+8x=0$ take $c=0$, though factoring out $2x$ is quicker there.