solve the quadratic equation by first factoring the perfect square trinomial on the left side. Then apply the square root properly.
x^2-10x+25=49
First off, let's factorize the left side of the equation, which is a perfect square trinomial.
x^2 - 10x + 25=49 can be factored as (x-5)^2=49.
Next step, we apply the square root property. Before we can apply the square root property, we need to isolate the perfect square trinomial on one side. The equation is already in this form.
Take the square root of both sides:
√((x-5)^2) = ±√49,
|x-5| = 7.
Since we are working with absolute values, there exist two possible solutions for x:
If it is positive:
x-5 = 7,
x = 7 + 5,
x = 12.
If it is negative:
x-5 = -7,
x = -7 + 5,
x = -2.
Therefore, the solutions to the equation are x = 12 and x = -2.
To solve the quadratic equation x^2 - 10x + 25 = 49 by factoring the perfect square trinomial, we need to rewrite the equation in the form (x - a)^2 = 0.
Subtract 49 from both sides of the equation:
x^2 - 10x + 25 - 49 = 0
Simplify:
x^2 - 10x - 24 = 0
Next, we need to find two numbers, let's call them b and c, such that b + c = -10 and b * c = -24, since the only way to factor a trinomial is to find two binomials whose sum and product are equal to the trinomial's coefficients.
In this case, the numbers -4 and 6 satisfy these conditions, since -4 + 6 = -10 and -4 * 6 = -24.
So we can factor the equation as follows:
(x - 4)(x + 6) = 0
Now, to solve for x, we set each factor equal to zero and solve for x:
x - 4 = 0 --> x = 4
x + 6 = 0 --> x = -6
Therefore, the solutions to the quadratic equation x^2 - 10x + 25 = 49 are x = 4 and x = -6.
To solve the quadratic equation x^2 - 10x + 25 = 49 by factoring and applying the square root, follow these steps:
Step 1: Rearrange the equation to have zero on one side: x^2 - 10x + 25 - 49 = 0
Step 2: Simplify the equation: x^2 - 10x - 24 = 0
Step 3: Factor the perfect square trinomial on the left side of the equation. In this case, the perfect square trinomial is (x - 5)^2. So, rewrite the equation by factoring it: (x - 5)^2 = 0
Step 4: Apply the square root to both sides of the equation to remove the square: √[(x - 5)^2] = √0
Step 5: Simplify the square root on the left side: x - 5 = 0
Step 6: Solve for x by isolating it on the left side, move -5 to the right side: x = 5
So, the solution to the quadratic equation x^2 - 10x + 25 = 49 is x = 5.