Solution: x = -3/5 and x = 3/5
To understand why please read following step by step solution.
STEP 1: Read, understand, identify important information, and translate keywords into mathematical symbols.
• 25 = 5^2; 9 = 3^2; x^2 = (x)(x) They are perfect square.
• The two terms of the binomial equation 25x^2 + -9 = 0 are perfect square.
• When factoring the difference of two perfect squares: x^2 – a^2 = (x+a)(x-a) .
STEP 2: Set up and solve the equation.
• 25 x^2 +– 9 = 0
(5x)^2 – (3)^2 = 0
(5x + 3) (5x -3) = 0
Solution A: Solution B:
5x + 3 = 0 5x - 3 = 0
-3 -3 +3 +3
5x = -3 5x = 3
5x/5 = -3/5 5x/5 = 3/5
x = -3/5 x = 3/5
STEP 3: Check your solutions.
• Verifying the factored form equation (5x + 3) (5x -3) = 0
(5x) (5x) + (5x) (-3) + (3) (5x) + (3) (-3) = 0 With the Distributive Prop.
25x^2 - 15x + 15x - 9=0 Using Distrib. Property & simplifying like terms,
25x^2-9=0 The original equation. So our factored form equation is right!
• Verifying the solutions: Just substitute x = -3/5 and x = 3/5 in the given equation 25x^2 - 9 = 0
For solution A: For solution B:
25(-3/5)^2 – 9 = 0 25(3/5)^2 – 9 = 0
25 (9/25) - 9 = 0 25 (9/25) - 9 = 0
(25/1)(9/25) - 9 = 0 (25/1)(9/25) -9 = 0
9 – 9 =0 9 – 9 =0 Both are identities, so the solutions A and B are true solutions.
STEP 4: Curiosities
• If you feel more comfortable by transforming the equation in the exact form: x^2 – a^2 = 0
25 x^2 – 9 = 0 Get rid of the 25
(1/25) (25 x^2 – 9 = 0)
(25/25)x^2 – 9/25 = 0/25
x^2 – 9/25 = 0
x^2 – 3^2/5^2 = 0 Applying exponent properties
x^2 – (3/5)^2 = 0 A perfect square binomial
(x + 3/5) (x - 3/5) = 0 And the solutions are x = -3/5 and x = 3/5
