The nature of roots of a quadratic equation can be found without actually finding the roots (α, β) of the equation. Here we shall learn more about how to find the nature of roots of a quadratic equation without actually finding the roots of the equation. The roots of a quadratic equation are usually represented to by the symbols alpha (α), and beta (β). Thus, by completing the squares, we were able to isolate x and obtain the two roots of the equation. This is good for us, because now we can take square roots to obtain: The left-hand side is now a perfect square: Now, we express the left-hand side as a perfect square, by introducing a new term (b/2a) 2 on both sides: To determine the roots of this equation, we proceed as follows: Proof of Quadratic FormulaĬonsider an arbitrary quadratic equation: ax 2 + bx + c = 0, a ≠ 0 This formula is also known as the Sridharacharya formula.Įxample: Let us find the roots of the same equation that was mentioned in the earlier section x 2 - 3x - 4 = 0 using the quadratic formula. Quadratic Formula: The roots of a quadratic equation ax 2 + bx + c = 0 are given by x = /2a. The positive sign and the negative sign can be alternatively used to obtain the two distinct roots of the equation. The two roots in the quadratic formula are presented as a single expression. There are certain quadratic equations that cannot be easily factorized, and here we can conveniently use this quadratic formula to find the roots in the quickest possible way. Quadratic formula is the simplest way to find the roots of a quadratic equation. Maximum and Minimum Value of Quadratic Expression Solving Quadratic Equations by Factorization Nature of Roots of the Quadratic Equation We shall learn more about the roots of a quadratic equation in the below content. These two solutions (values of x) are also called the roots of the quadratic equations and are designated as (α, β). Quadratic equations have maximum of two solutions, which can be real or complex numbers. Did you know that when a rocket is launched, its path is described by a quadratic equation? Further, a quadratic equation has numerous applications in physics, engineering, astronomy, etc. In other words, a quadratic equation is an “equation of degree 2.” There are many scenarios where a quadratic equation is used. The term "quadratic" comes from the Latin word "quadratus" meaning square, which refers to the fact that the variable x is squared in the equation. This is, of course, redundant.Quadratic equations are second-degree algebraic expressions and are of the form ax 2 + bx + c = 0. Note that if b = 0, then (b/2)² = 0, and so we would add 0 to both sides of the equations. If b=0, then you may skip Steps 2, 3, and 4 and go from x² = -c (Step 1) directly to x = ±√|c| (Step 5).īecause in Step 2, we take b and perform some arithmetic operations on it, which gives us the number with which we will 'complete the square.' Now just go ahead with the steps we explained in the example above. Dividing either side by 2, we obtainĪnd so the coefficient in front of x² is equal to 1. transform the problem to some you've already solved□ What does that mean in our context? Just divide your equation by a!īy completing the square. What do you do if you are asked to solve a quadratic equation where a ≠ 1? Just apply one of the most frequently used problem-solving techniques in math, namely. In the example above, it is important that the coefficient in front of x² is equal to 1. This means we've determined the points where the parabola y = x² + 6x - 7 intersects the x-axis. Hence, we've found the solutions of x² + 6x - 7 = 0. Recall the short multiplication formula, (p + q)² = p² + 2pq + q², and note that we may apply it 'backwards' to the left-hand side of our equation (with p = x and q = 3). Now it's time to complete the square! Take one-half of the coefficient in front of x and square it:Īdd the number computed in Step 2 to both sides of the equation: We break the process into several simple steps so that nobody gets overwhelmed by the formula for completing the square:Īdd 7 to either side of the equation so that the left-hand side contains only terms with x:
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