![]() ![]() Solving quadratic equations by graphing.Solving quadratic equations by completing the square.Solving quadratic equations by factoring.But how to find them if they are not given? There are different ways of solving quadratic equations. ![]() Thus, x = 1 and x = 2 are the roots of x 2 - 3x + 2 = 0. ![]() For example, one can easily see that x = 1 and x = 2 satisfy the quadratic equation x 2 - 3x + 2 = 0 (you can substitute each of the values in this equation and verify). Since the degree of the quadratic equation is 2, it can have a maximum of 2 roots. The value(s) that satisfy the quadratic equation is known as its roots (or) solutions (or) zeros. Solving quadratic equations means finding a value (or) values of variable which satisfy the equation. Solving Quadratic Equations by Quadratic Formula Solving Quadratic Equations by Completing Square Let us learn all the methods in detail here along with a few solved examples. But the most popular method is solving quadratic equations by factoring. There are different methods used to solve quadratic equations. We know that any value(s) of x that satisfies the equation is known as a solution (or) root of the equation and the process of finding the values of x which satisfy the equation ax 2 + bx + c = 0 is known as solving quadratic equations. The standard form of a quadratic equation is given by the equation ax 2 + bx + c = 0, where a ≠ 0. It means the quadratic equation has a variable raised to 2 as the greatest power term. The word "quadratic" is originated from the word "quad" and its meaning is "square". #3(x-7/6)^2=13/12#.Before going to learn about solving quadratic equations, let us recall a few facts about quadratic equations. (Note the signs in the middle and the return of the #7/6# that we squared earlier.) Keeping the perfect square together, we re-write this as: To avoid changing the number (not just the way it's written) we'll also subtracting. To make the expression in parehtheses inc lude a complete square, we need to add #(7/6)^2# which is #49/36#. The middle term is #7/3x# Recall that the middle term of #(x+n)^2# is #2nx#. Now we will complete the square inside the parentheses. But, since #7# is not divisible by #3#, we just wrote #7/3#. Which is true if and only ifĭo you see what we did there? We factored out a #3#. (You should probably read the first one first.) I'll post another (more challenging) example too. The solution set to the first equation is: #. So the first equation is equivalent toĪnd the last equation above is satisfied exactly when: Solve: #x^2+6x-16=0# (by completing the square)Įach of the following equations is equivalent (has exactly the same solutions) as the lines before it. Solving an equation by completing the square: ![]() We write #x^2+6x+9-9# If we group it this way: #(x^2+6x+9)-9# then we have a perfect square minus #9# That doesn't change the value of #x^2+6x#, but it does change the way it's written. Of course you can's just add a number to an expression without changing the value of the expression, so if we want to keep the same value we'll have to make up for adding #9#. We can figure out what to use for #n# by realizing that the #6x# in the middle need to be #2nx#. To make it complete, we'd need to add #n^2# to the expression. Notice: the sign on the middle term matches the sign in the middle of the binomial on the left AND the last term is positive in both.Īlso notice that if we allow #n# to be negative, we only need to write and think about #(x+n)^2=x^2+2nx+n^2# (The sign in the midde will match the sign of #n#.)Īn expression like #x^2+6x# may be thought of as an "incomplete" square. The square of an expression of the form #x+n# or #x-n# is: ![]()
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