Although the standard form (Eq. 1) of the quadratic expression doesn’t relate to the vertex or x-intercepts of the parabola in the same way that the vertex or factored forms do respectively, they are still useful ways of expressing the equation, as they clearly display the equation in a simple format with all pertinent information present, and consistent with the format of other polynomial functions. Additionally, many physical phenomena in science are best described as quadratic equations in this format.

Eq. 1: \(y=ax^2+bx+c\)

One way to analyze these equations directly is to use the quadratic formula (Eq. 2).

Eq. 2: \(x={-b\pm\sqrt{b^2-4ac}\over2a}\)

The quadratic formula is an equation that can identify the roots (where the parabola crosses the x-axis) of the equation. This equation includes the ± symbol, which indicates that you have two separate equations which have an addition and a subtraction form. This means that there are two solutions to the quadratic formula, reflecting the fact that quadratic expressions contain two different roots where the line crosses the 0 axis. Taking an example of a simple quadratic equation in the standard form (Eq. 3), you should be able to see that this equation factors to give +2 and +3, meaning that the roots are -2 and -3.

Eq. 3: \(y=x^2+5x+6\)

We can verify the roots of this equation are -2 and -3 by putting the equation into the quadratic formula (Eq. 2), substituting the a, b, and c values for the appropriate values in the quadratic formula (Eq. 4). This is a complex equation, so remember to follow the correct order of operations to solve it.

Eq. 4: \(x={-5\pm\sqrt{5^2-4\times1\times6}\over2\times1}\)

The ± symbol indicates that this equation have 2 solutions, a solution made by adding the root (Eq. 5) to the numerator of the division, and a solution made by subtracting the root (Eq. 6).

Eq. 5: \(x={-5+\sqrt{5^2-4\times1\times6}\over2\times1}=-2\)

Eq. 6: \(x={-5-\sqrt{5^2-4\times1\times6}\over2\times1}=-3\)

This gives our 2 roots of our quadratic expression of -2 and -3. This example is simple, as you could have also got these solutions by factoring. However there are other types of quadratic expression that aren’t so easy to solve.

Expressions That Can’t Be Factored

Obviously if a factored form of a quadratic expression exists then it is possible to factor. But just how difficult do you think it would be to factor a quadratic expression that has decimals, and likely has factors which are decimals (Eq. 7).

Eq. 7: \(y=x^2+5.3x+6.3\)

If instead of trying to factor this expression, we can put the a, b, and c values into the quadratic formula (Eq. 8). Solving this (Eq. 9 and Eq. 10) gives us our 2 roots of -1.8 and -3.5. These would be the solutions to the factored form as well (Eq. 11).

Eq. 8: \(x={-5.3\pm\sqrt{5.3^2-4\times1\times6.3}\over2\times1}\)

Eq. 9: \(x={-5.3+\sqrt{5.3^2-4\times1\times6.3}\over2\times1}=-1.8\)

Eq. 10: \(x={-5.3-\sqrt{5.3^2-4\times1\times6.3}\over2\times1}=-3.5\)

Eq. 11: \(y=(x+1.8)(x+3.5)\)

Finding The Intersection With a Given y Value

If we have an expression with a given result in place of the y value, we are looking for x values that correspond to that given value (Eq. 12). We can use the quadratic formula to solve this problem.

Eq. 12: \(40=2x^2+3x+4\)

First of all, we can rearrange this equation so that the left side of the equation equals 0, by subtracting 40 from both sides (Eq. 13).

Eq. 13: \(0=2x^2+3x-36\)

This is then a quadratic expression in the standard form, and equalling 0, which we can then add the a, b, and c values into our quadratic formula (Eq. 14).

Eq. 14: \(x={-3\pm\sqrt{3^2-4\times2\times36}\over2\times2}\)

We can then solve this quadratic formula for the + and – operations to give the two x values that where the parabola equals the y value of 40 (Eq. 15 and Eq. 16). We can see here that the x coordinates are -5.06 and 3.56.

Eq. 15: \(x={-3+\sqrt{3^2-4\times2\times36}\over2\times2}=3.56\)

Eq. 16: \(x={-3-\sqrt{3^2-4\times2\times36}\over2\times2}=-5.06\)

If we plot the parabola produced by the quadratic expression, we can verify this answer (Figure 1).

Figure 1: The parabola formed by the function y=2x²+3x+4. To find the intersection with y=40 (green line), this equation can be rearranged to 0=2x²+3x-36 and using the quadratic formula, we find the x values where y=40 are 3.56 and -5.06 (red lines).

Verifying if an Equation can be Factored

If you look at the quadratic formula, you’ll notice that there is a square root in it (Eq. 17). Square roots of negative numbers have no solution, since no number squared will make a negative number. However, the numbers in this part of the quadratic formula can make a negative number, so the quadratic formula can contain a square root of a negative number. How does this work?

Eq. 17: \(\sqrt{b^2-4ac}\)

What this means is that this part of the quadratic formula can be used to identify if there are any roots for this equation at all. To do this you can follow these rules (Table 1).

Table 1: Using the square root part of the quadratic formula with different equations to see how many roots are present. Where the contents of the square root make a result greater than 0, the expression has 2 roots. When the contents of the square root total 0, the expression only has one root, whereas a value less than 0 will be an expression that has no roots.