Create New Account
Create New Account

Sign in to Innings2

or

New Account     Reset Password     

Quadratic Equations

    Nature of Roots

Powered by Innings 2

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

10th class > Quadratic Equations > Nature of Roots

Nature of Roots

Completing the square can be tricky, and it is easy to make mistakes along the way.

Let's follow the steps when completing the square, but use a, b and c as coefficients for the quadratic equation, rather than actual numbers:

Completing the square is long and complicated, and it is easy to make mistakes. Luckily, there is a shortcut that makes it a lot simpler!

To find it, we need to repeat the process of completing the square, but leaving the coefficients as a, b and c.

Lets start with a quadratic equation of the form.

ax2+bx+c=0.

To make the first term a perfect square, we have to multiply the entire equation by a. To avoid fractions, we also multiply by 4. This gives;

4a2x2 + 4abx + 4ac = 0.

Now apply the box method:

2axb
2ax 4ax2 2axb
b 2axbb2

These steps were ugly, painful, and you don't need to remember them (even though it was just the same as completing the square, just with variables). The result, however, was worth it: a single equation that tells us the solutions of any quadratic equation. It is often called the Formula:

x=−b±b2−4ac2a

If b2-4ac > 0, we get two distinct real roots −b2a + x=−b±b2−4ac2a and −b2a - x=−b±b2−4ac2a

If b2-4ac = 0, then x = −b2a+-0, i.e., x=−b2a. So, the roots of the equation ax2 + bx + c = are both −b2a.

To solve a quadratic equation, we just have to replace a, b and c with the actual numbers in our case, and then simplify the fraction.

Some curricula feel it is important to notice that the formula x=−b±b2−4ac2a represents two symmetrical values about the middle point x=−b2a.

The Discriminant

One particularly important part of the quadratic equation is the term under the square root, which is called the discriminant. Depending on the value of b2−4ac, you can tell a lot about the solutions of a quadratic equation, without ever actually solving it:

  • If b2−4ac<0, the quadratic equation has no solutions, because we cannot take square roots of negative numbers. (More on that later…)
  • If b2−4ac=0, the quadratic equation has one solution. Zero is the only number with just one square root, because +0=−0.
  • If b2−4ac>0, the quadratic equation has two solutions like before, one when evaluating the quadratic formula with +, and one when evaluating it with –.

(i) two real roots, if b2−4ac>0,

(ii) two real roots, if b2−4ac=0,

(iii)no roots, if b2−4ac<0.

Find the discriminant of the quadratic equation 2x2 – 4x + 3 = 0, and hence find the nature of its roots.

The given equation is of the form ax2+ bx + c = 0, where a = , b = and c = .
Therefore, the discriminant b2-4ac = −42 - 4 × 2 × 3 = - =
Hence, -8 < 0 So, the given equation has no roots.

A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

calculate the distance from two gates

  • Let the distance of the pole from the gate B be x m, i.e., BP = x m. Now the difference of the distances of the pole from the two gates = AP – BP (or, BP – AP) = 7 m. Therefore, AP = (x + 7) m.
  • Now, AB = m, and since AB is a diameter,
  • Therefore, AP2 + BP2 = AB2 (By Pythagoras theorem)
  • Split the equation is x2 + x + + x2 =
  • Add the terms and variables = x2 + x - = 0
  • So, the distance ‘x’ of the pole from gate B satisfies the equation x2 + x - = 0
  • So, it would be possible to place the pole if this equation has real roots. To see if this is so or not, let us consider its discriminant. The discriminant is b2 - 4ac = 72 - 4 × ×
  • substitute the terms in the equation then we get the answer is
  • Solving the quadratic equation x2 + 7x – 60 = 0, by the quadratic formula, x=−b±b2−4ac2a we get x=−7±2892 =
  • Therefore, x = 5 or

Find the discriminant of the equation 3x2-2x + 13 = 0 = 0 and hence find the nature of its roots. Find them, if they are real.

find the nature of roots

  • Here a = , b = and c =
  • Therefore, discriminant b2- 4ac = - × 3 × 13
  • Calculate the terms are - =
  • Hence, the given quadratic equation has two equal roots. The roots are −b2a,−b2a = ,.
  • Therefore ,.

Solving Quadratic Equations

We now saw multiple different ways to solve quadratic equations, all of which have advantages and disadvantages:

  • Basic Algebra This is the easiest way, but it only works for quadratic equations that don't contain an x-term.

  • Factoring Also quite simply, but it takes some guesswork and it doesn't always work.

  • Completing the Square Very long and complicated. It is easy to make mistakes. In addition to finding the solutions of an equation, it also tells us the vertex of the corresponding parabola.

  • Quadratic Formula Straightforward formula that always work, but it sometimes feels like "magic" and it is easy to forget why and how it works.