Quadratic Formula: How to Solve Quadratic Equations

The quadratic formula says x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. In the standard equation ax squared plus bx plus c equals zero, a is the coefficient of x squared, b is the coefficient of x, and c is the constant term.

Before substituting numbers, rewrite the equation so one side equals zero and identify a, b, and c carefully. Signs matter. If the equation is 2x squared minus 3x minus 2 equals zero, then a is 2, b is negative 3, and c is negative 2. Writing those values separately can prevent a small sign error from spreading through the calculation.

This preparation is a big part of using the quadratic formula well. The formula itself is fixed, but most mistakes happen while reading the coefficients, squaring b, or handling the negative sign in front of b.

Take x squared minus 5x plus 6 equals zero. Here a is 1, b is negative 5, and c is 6. Substituting into the quadratic formula gives the negative of negative 5, plus or minus the square root of 25 minus 24, all divided by 2. That simplifies to 5 plus or minus 1, divided by 2.

The two choices created by plus or minus give two solutions. Using plus gives 6 divided by 2, so x equals 3. Using minus gives 4 divided by 2, so x equals 2. These are the same roots you would find by factoring the quadratic as x minus 2 times x minus 3.

The example shows why the quadratic formula is dependable. Factoring can be quick when the numbers are friendly, but the formula uses the same process every time and works even when the roots are not neat integers.

The expression under the square root is called the discriminant, and it gives useful information before you finish solving:

• •A positive discriminant gives two distinct real solutions.
• •A zero discriminant gives one repeated real solution.
• •A negative discriminant gives no real solutions, though complex solutions exist.
• •A perfect-square discriminant often leads to simple rational roots.

A useful habit with the quadratic formula is to check your solutions in the original equation. Substitute each value of x back into the left side and see whether the result equals zero. This catches arithmetic mistakes and confirms that both branches of the plus-or-minus sign were handled. You can also compare methods. If a quadratic factors easily, solve it by factoring and then use the formula as a check. If the two methods disagree, revisit the signs and coefficients. Over time, this comparison builds confidence in choosing a method rather than treating every quadratic equation as a completely new problem.

The quadratic formula is not a rule you have to take on trust. It comes from a method called completing the square. Start with ax squared plus bx plus c equals zero and divide every term by a. Move the constant across, leaving x squared plus b over a times x equals negative c over a. Now add the square of half the x coefficient, which is b squared over 4a squared, to both sides. The left side becomes a perfect square, x plus b over 2a, all squared, while the right side simplifies to b squared minus 4ac, all over 4a squared.

Take the square root of both sides, remembering the plus or minus, and subtract b over 2a. The result is exactly the formula. The derivation also explains the graph of a quadratic. The term negative b over 2a is the x coordinate of the vertex, the turning point of the parabola, which lies halfway between the two roots. The plus or minus part tells you how far each root sits on either side of that line of symmetry. When the discriminant is negative, there is no real distance to step, so the parabola never crosses the x axis.

In real problems the formula often gives two answers, and only one may fit the situation. Suppose a rectangular garden is 3 metres longer than it is wide and has an area of 30 square metres. If the width is w, then w times w plus 3 equals 30, which rearranges to w squared plus 3w minus 30 equals zero. Here a is 1, b is 3 and c is negative 30, so the discriminant is 9 plus 120, which is 129. Its square root is about 11.36, giving w as roughly 4.18 or negative 7.18.

A width cannot be negative, so the garden is about 4.18 metres wide and 7.18 metres long, and multiplying those does give about 30. A thrown ball works the same way. If its height in metres after t seconds is 2 plus 15t minus 4.9t squared, it lands when that expression equals zero. The formula gives t as about 3.19 seconds or about negative 0.13 seconds. The negative root is not nonsense. It marks where the same curve would have met the ground before the throw, outside the stretch of time the model is meant to describe.

People were solving quadratic problems long before the formula was written with letters. Old Babylonian clay tablets, written nearly 4,000 years ago, contain problems such as this: the area of a square added to its side gives three quarters, so what is the side? The scribes solved them with step-by-step recipes in words that match completing the square, and reached the answer of one half without any symbols at all. They worked in base 60 and wanted practical results, so their problems were set up to give positive answers that could stand for real lengths and areas.

In the ninth century the Baghdad scholar al-Khwarizmi, whose book gave us the word algebra, explained the method with a picture. His example was a square and ten roots equal to thirty-nine, which we would write as x squared plus 10x equals 39. Draw a square of side x and attach a 5 by x rectangle to each of two sides, giving a shape of area 39. The missing corner is a 5 by 5 square of area 25, so the completed square has area 64 and side 8. Taking away 5 leaves x equals 3. Negative answers were not accepted, so the second root, negative 13, went unmentioned.

The quadratic formula is a reliable tool for solving equations with an x squared term. Put the equation in standard form, identify a, b, and c with their signs, substitute carefully, and use both parts of the plus or minus. The discriminant gives you an early clue about the kinds of solutions you should expect.