Question

Prove that x=\frac{5}{4}x=45 and x = −3 are the roots of the equation 4x2 + 7x − 15 = 0.​

Answer 1

Answer:

This approach to solving equations is based on the fact that if the product of two quantities is zero, then at least one of the quantities must be zero. In other words, if a*b = 0, then either a = 0, or b = 0, or both. For more on factoring polynomials, see the review section P.3 (p.26) of the text.

Example 1.

2x2 - 5x - 12 = 0.

(2x + 3)(x - 4) = 0.

2x + 3 = 0 or x - 4 = 0.

x = -3/2, or x = 4.

Square Root Principle

If x2 = k, then x = ± sqrt(k).

Example 2.

x2 - 9 = 0.

x2 = 9.

x = 3, or x = -3.

Example 3.

Example 4.

x2 + 7 = 0.

x2 = -7.

x = ± .

Note that = = , so the solutions are

x = ± , two complex numbers.

Completing the Square

The idea behind completing the square is to rewrite the equation in a form that allows us to apply the square root principle.

Example 5.

x2 +6x - 1 = 0.

x2 +6x = 1.

x2 +6x + 9 = 1 + 9.

The 9 added to both sides came from squaring half the coefficient of x, (6/2)2 = 9. The reason for choosing this value is that now the left hand side of the equation is the square of a binomial (two term polynomial). That is why this procedure is called completing the square. [ The interested reader can see that this is true by considering (x + a)2 = x2 + 2ax + a2. To get "a" one need only divide the x-coefficient by 2. Thus, to complete the square for x2 + 2ax, one has to add a2.]

Step-by-step explanation:

hope it will help you