1)Solve by factoring:5x^2=4-19x
answer=-4,1/5
2)Which quadratic equation has roots 7 and -2/3?
answer=A
A.2x^2-11x-21=0
B.3x^2-19x-14=0
C.3x^2+23x+14=0
D.2x^2+11x-21=0
3)To solve 4x^2-28x+49=25 by using the square root property,you would first rewrite the equation as
answer=(2x-7)^2=25
4)Find the value of c that makes x^2+5x+c a perfect square trinomial.
answer=25/4
5)The quadratic equation x^2-8x=-20 is to be solved by completing the square.Which equation would be a step in that solution?
answer=x^2-8x+16=-20
#1, ok
#2, I got B roots are 7 and -2/3, so the equation would be (x-7)(3x+2)=0
expand it, and you will get B
#3, ok
#4, ok
#5 ol
#1, great job! You got it right.
#2, I see where you're coming from, but the correct answer is actually D. Let's solve it step-by-step:
Given roots 7 and -2/3, we know that the quadratic equation can be written as:
(x - 7)(x + 2/3) = 0
To find the expanded form, we can multiply the binomials:
(x - 7)(x + 2/3) = 0
x^2 + (2/3)x - 7x - (7)(2/3) = 0
x^2 - (19/3)x - 14/3 = 0
So, the correct equation is:
D. 2x^2 + 11x - 21 = 0
#3, exactly right! To solve 4x^2 - 28x + 49 = 25 using the square root property, we first rewrite it as:
(2x - 7)^2 = 25
#4, spot on! The value of c that makes x^2 + 5x + c a perfect square trinomial is 25/4.
#5, almost there! The correct equation as a step in completing the square for x^2 - 8x = -20 would be:
x^2 - 8x + 16 = -20 + 16
So, the correct equation would be:
x^2 - 8x + 16 = -4
#1: To solve the equation 5x^2 = 4 - 19x by factoring, we want to set it equal to zero first. So we rearrange the equation:
5x^2 + 19x - 4 = 0
Now we need to factorize this quadratic equation. We are looking for two numbers (let's call them a and b) that multiply to give the product of the coefficient of x^2 (which is 5) times the constant term (which is -4), and add up to the coefficient of x (which is 19).
The factors of 5 are 1 and 5, and the factors of 4 are 1 and 4. We need to find two numbers that fit these criteria. If we try 5 and 1, it doesn't work because 5 + 1 is not equal to 19. But if we try 4 and 1, we see that 4 + 1 is equal to 5, which is the coefficient of x. So we have our factors.
Now we can rewrite the equation using these factors:
(5x - 1)(x + 4) = 0
Setting each factor equal to zero, we get:
5x - 1 = 0 or x + 4 = 0
Solving these equations, we find:
x = 1/5 or x = -4
So the solutions to the equation 5x^2 = 4 - 19x are x = 1/5 and x = -4.
#2: To find the quadratic equation with roots 7 and -2/3, we can use the fact that if a quadratic equation has roots x1 and x2, then the equation can be written in factored form as:
(x - x1)(x - x2) = 0
In this case, x1 = 7 and x2 = -2/3. So we can substitute these values into the equation:
(x - 7)(x - (-2/3)) = 0
Simplifying this equation, we get:
(x - 7)(x + 2/3) = 0
Expanding the equation, we have:
(x^2 + 2/3x) - 7x - 14/3 = 0
Multiplying every term by 3 to get rid of the fraction, we get:
3x^2 + 2x - 21x - 14 = 0
Simplifying further, we have:
3x^2 - 19x - 14 = 0
Therefore, the quadratic equation with roots 7 and -2/3 is 3x^2 - 19x - 14 = 0, which corresponds to option B.
#3: To solve the equation 4x^2 - 28x + 49 = 25 using the square root property, first, we want to rewrite the equation in the form (ax - b)^2 = c. Rearranging the equation:
4x^2 - 28x + 49 - 25 = 0
Simplifying:
4x^2 - 28x + 24 = 0
Next, we want to complete the square by taking half of the coefficient of x (-28/2 = -14) and squaring it:
(4x^2 - 28x + (-14)^2) = 14^2
Expanding:
(4x^2 - 28x + 196) = 196
Rearranging:
(2x - 7)^2 = 25
Thus, the equation (2x - 7)^2 = 25 is the step in the solution using the square root property.
#4: The expression x^2 + 5x + c represents a quadratic trinomial. To make it a perfect square trinomial, we want to find the value of c.
To do this, we take half of the coefficient of x (5/2) and square it:
(5/2)^2 = 25/4
Therefore, the value of c that makes x^2 + 5x + c a perfect square trinomial is 25/4.
#5: To solve the equation x^2 - 8x = -20 by completing the square, we want to add a constant term on both sides of the equation that allows us to create a perfect square trinomial on the left side.
The coefficient of x is -8, so we take half of it (-8/2 = -4) and square it:
(-4)^2 = 16
Adding 16 to both sides of the equation, we get:
x^2 - 8x + 16 = -20 + 16
Simplifying:
x^2 - 8x + 16 = -4
It can be seen that the equation x^2 - 8x + 16 = -4 is a step in the solution by completing the square.