Forming and Transforming Equations

Transcript

0:00Introduction

Hello everyone! Now, one small challenge before we start. Take the quadratic two x squared, plus three x, minus one, equals zero, and call its roots alpha and beta. Now I want a new quadratic whose roots are alpha plus two, and beta plus two. Most students find alpha and beta with the quadratic formula, add two, then build the new equation. Lots of work! Can you do it without finding the roots at all? Toppers do it in one line. And look, JEE does ask such questions. So today: forming equations, and transforming them. Let's begin.

0:39Forming an equation from its roots

So, first, forming an equation. Suppose I give you two numbers, alpha and beta, the Greek letters on screen. You have to build a quadratic whose roots are exactly alpha and beta. Which equation will you write? Yes, simple! x minus alpha, into x minus beta, equals zero. Put x equals alpha, and the first bracket becomes zero. Put x equals beta, and the second bracket becomes zero. So both are roots, right? Now open the brackets. x squared, minus, alpha plus beta, into x, plus alpha beta, equals zero. The coefficient of x squared here is one; such a quadratic is called monic, okay? In words: x squared, minus sum into x, plus product, equals zero. Careful with the signs: minus the sum, plus the product. In a hurry, students write plus the sum and lose marks. Clear?

1:36Transforming the roots: the one-line rule

Now, transforming. Take any quadratic, f of x, equal to a x squared plus b x plus c, equals zero, with roots alpha and beta. Here a, b and c are just the coefficients, okay? I want new roots alpha plus h and beta plus h, so each root moves up by some number h. Roots go up by h, so we simply replace x by x plus h. Right? No, no, careful! Let us actually check. Call the new root y, so y equals alpha plus h. This double arrow means, so, or implies: alpha equals y minus h. Now, alpha is a root of f, so f of alpha is zero. Put alpha equals y minus h, and we get f of y minus h equals zero. Minus h, not plus h! So, for roots alpha plus h and beta plus h, the equation is f of x minus h equals zero. Roots go up, the replacement goes the opposite way. This is the JEE trap. Got it? And here is the whole method in general. If the new root is y equals g of alpha, solve it for alpha: alpha equals g inverse of y. The small minus one on top means the inverse, okay? Then substitute into f. That's all there is to it.

2:57Multiply, negate, flip the roots

Alright, now the same idea for other changes. Roots multiplied by a number k: k alpha and k beta. Same logic as before. What should replace x? Yes, x over k! The new root y is k alpha, so alpha equals y over k. The roots get multiplied, so the replacement divides. Opposite way again, right? Put it in: a x squared over k squared, plus b x over k, plus c, equals zero. Multiply through by k squared: a x squared, plus b k x, plus c k squared, equals zero. So the middle coefficient gets one k, and the constant gets k squared. Roots minus alpha and minus beta? Replace x by minus x. The x squared term does not change, and only the middle term flips its sign: a x squared, minus b x, plus c, equals zero. Nice and simple. And now my favourite: reciprocal roots, one over alpha and one over beta. Put one over x in place of x, then multiply everything by x squared. What do you get? c x squared, plus b x, plus a, equals zero! Look, the coefficients just come in reverse order, okay? Almost no work in the exam.

4:14Topper's trick: the squaring method

Last one: squared roots, alpha squared and beta squared. The rule says: replace x by the square root of x, this tick-shaped root sign. But square roots inside an equation are messy, right? So toppers use a cleaner way. Here is the topper's trick, the squaring method. Start from a x squared plus b x plus c equals zero, and keep the b x term alone on one side: a x squared plus c, equals minus b x. Now look. The left side has only x squared, but the right side has a plain x. How do we turn that plain x into x squared too? Square both sides, exactly! Then, a x squared plus c, whole squared, equals b squared x squared. Now every x appears only as x squared. And x squared is exactly what our new roots are made of, isn't it? So replace x squared by x: a x plus c, whole squared, equals b squared x. That is the equation with roots alpha squared and beta squared. See, this is much faster than finding alpha squared plus beta squared, and alpha squared beta squared. Remember this one, okay? We will use it in a minute.

5:27Example 1: roots shifted by 2, then reciprocal roots

Now, remember our challenge from the start? Let's finish it. Alpha and beta are the roots of two x squared plus three x minus one equals zero. Find the equation with roots alpha plus two and beta plus two, and the one with roots one over alpha and one over beta. Pause the video, and try it first. Done? Okay, let's check. The roots go up by two. So, quick: do we replace x by x plus two, or by x minus two? x minus two, that's right! You remembered the trap. So: two into x minus two, whole squared, plus three into x minus two, minus one, equals zero. Open the brackets: two x squared minus eight x plus eight, then plus three x minus six, then minus one. Collect the terms: two x squared, minus five x, plus one, equals zero. One line, and we never found alpha or beta. That is the topper's way from our hook! Now the second part, reciprocal roots, one over alpha and one over beta. Which shortcut did we learn for this? Yes, reverse the coefficients! Two, three, minus one becomes minus one, three, two: minus x squared, plus three x, plus two, equals zero. Change all the signs, and that is x squared, minus three x, minus two, equals zero. Clear?

6:57Example 2: equation with squared roots by the squaring trick

Now one more, with squared roots. Alpha and beta are the roots of x squared minus three x plus two equals zero. Find the equation whose roots are alpha squared and beta squared. Pause and try. Hint: use the squaring method. Done? Okay, let's check. Keep the x term alone on one side: x squared plus two equals three x. Square both sides. x squared plus two, whole squared, is x to the power four, plus four x squared, plus four. And three x, squared, is nine x squared. Bring it across: x to the power four, minus five x squared, plus four, equals zero. So this is our answer, right? But wait, it has x to the power four. Is that okay? No, no! We need a quadratic. Every x here comes as x squared, so replace x squared by x: x squared, minus five x, plus four, equals zero. Forgetting this last step is the classic slip, okay? And here a check is easy. The old roots were one and two, and the arrow takes us to the new roots, one and four, their squares. This tick mark means it matches. Absolutely correct!

8:12Quick revision: form, shift, reverse and square the roots

So, quick revision. Forming: x squared, minus sum into x, plus product, equals zero. Transforming: solve y equals g of alpha for alpha, and substitute into f. Roots plus h means x minus h; reciprocal roots means reverse the coefficients; and squared roots means the squaring trick. Now go and try the practice problems on this page. You will find them easy. All the best!