Two questions actually. Please only respond with step-by-step answers. Feel free to change the numbers, I just need to know how to do it...
1.) Factor Completely:
4a^2 + 81 - 36a
2.) In a right triangle, find the length of the side not given. b= 2 c= sqrt. 13
1.) 4a^2 -36a + 81
What are factors of 81? (3 * 3 * 3 * 3)
What are factors of 4? (2 * 2)
What combination of these factors (or the whole number) can give you -36 for the middle term?
2.) Pythagorean theorem
a^2 + b^2 = c^2
a^2 + 4 = 13
Can you work it from here?
Sure! Here are step-by-step instructions for both questions:
1.) Factor Completely:
To factor the expression 4a^2 + 81 - 36a, follow these steps:
Step 1: Write down the expression: 4a^2 + 81 - 36a
Step 2: Look for any common factors. In this case, there are none.
Step 3: Identify if the expression is a quadratic trinomial or a perfect square trinomial. In this case, it is a quadratic trinomial.
Step 4: Use the quadratic formula to find the factors. The quadratic formula is given by:
a) x = (-b + sqrt(b^2 - 4ac)) / (2a)
b) x = (-b - sqrt(b^2 - 4ac)) / (2a)
where a, b, and c are the coefficients in the quadratic equation ax^2 + bx + c = 0.
In our case, the quadratic trinomial is 4a^2 - 36a + 81, so a = 4, b = -36, and c = 81.
Step 5: Plug these values into the quadratic formula and solve for x. This will give you the factors of the quadratic trinomial.
2.) Finding the length of the missing side in a right triangle:
To find the length of the missing side in a right triangle when given two other sides, follow these steps:
Step 1: Write down the given side lengths. In this case, b = 2 and c = sqrt(13).
Step 2: Identify the sides of the right triangle. The three sides of a right triangle are typically labeled as a, b, and c, where c is the hypotenuse (the side opposite the right angle).
Step 3: Use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (a and b) is equal to the square of the hypotenuse (c^2 = a^2 + b^2).
Step 4: Substitute the given values into the Pythagorean theorem equation: (sqrt(13))^2 = 2^2 + a^2.
Step 5: Simplify and solve the equation: 13 = 4 + a^2.
Step 6: Subtract 4 from both sides: 9 = a^2.
Step 7: Take the square root of both sides: sqrt(9) = sqrt(a^2).
Step 8: Simplify: 3 = a.
Therefore, the length of the missing side (a) in the right triangle is 3.