a) expand and simplify: (4x-3)(x+1)
i get 4x^2+x-3 ?
b) factorise: 2x^2 + 13x + 15
c) find y-int, x-int and vertex of:
y= x^2-3x-4
i get y-int = -4, but loose it from there :(
A. correct
B. look for two numbers m,n such that mn=2*15=30, and m+n=13.
m and n are therefore 10 and 3.
2x²+10x + 3x+15
=2x(x+5) + 3(x+5)
=(2x+3)(x+5)
C. y-intercept: correct.
X-intercept=zeroes.
Factorize y=x²-3x-4=(x+1)(x-4) to get the zeroes.
The vertex is located at x=-b/2a=-(-3)/2=1.5
and corresponding y is f(1.5)=-6.25
The vertex can also be found by completing the square.
a) To expand and simplify the expression (4x-3)(x+1), you can use the distributive property. Multiply each term in the first parentheses by each term in the second parentheses:
(4x)(x) + (4x)(1) + (-3)(x) + (-3)(1)
This simplifies to:
4x^2 + 4x - 3x - 3
Combining like terms, we have:
4x^2 + x - 3
So you are correct with your answer of 4x^2 + x - 3.
b) To factorize the quadratic expression 2x^2 + 13x + 15, you need to find two binomials that multiply together to give you the original expression.
Start by looking at the factors of the coefficient of x^2, which is 2. The factors of 2 are 1 and 2. Next, you need to find two numbers that multiply together to give you the product of the coefficient of x^2 and the constant term, which is 2x^2 * 15 = 30. The numbers that multiply to give you 30 are 1 and 30, 2 and 15, or 3 and 10.
Now, you need to find the combination of these factors that add up to the coefficient of x, which is 13. In this case, the combination that works is 2 and 15.
So, you can split the middle term of 13x into 2x and 15x, and rewrite the expression as follows:
2x^2 + 2x + 15x + 15
Now, factor by grouping:
(2x^2 + 2x) + (15x + 15)
Factor out the greatest common factor from each group:
2x(x + 1) + 15(x + 1)
Combine the factors:
(2x + 15)(x + 1)
Therefore, the factored form of 2x^2 + 13x + 15 is (2x + 15)(x + 1).
c) To find the y-intercept, x-intercept, and vertex of the quadratic function y = x^2 - 3x - 4, you can use a few different methods.
1. Y-intercept:
The y-intercept is the value of y when x is 0. So substitute x = 0 into the equation:
y = (0)^2 - 3(0) - 4
y = -4
Therefore, the y-intercept is -4.
2. X-intercepts:
The x-intercepts are the values of x when y is 0. Set the equation to 0 and solve for x:
x^2 - 3x - 4 = 0
To factorize this quadratic equation, you need to find two numbers that multiply to give you the constant term (-4) and add up to give you the coefficient of x (-3). In this case, the numbers -4 and 1 fit these criteria.
(x - 4)(x + 1) = 0
Setting each factor equal to zero gives:
x - 4 = 0 => x = 4
x + 1 = 0 => x = -1
Therefore, the x-intercepts are x = 4 and x = -1.
3. Vertex:
The vertex of a quadratic function can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation (in the form ax^2 + bx + c).
In the equation y = x^2 - 3x - 4, a = 1, b = -3, and c = -4.
x = -(-3)/(2*1)
x = 3/2
To find y, substitute the value of x into the equation:
y = (3/2)^2 - 3(3/2) - 4
y = 9/4 - 9/2 - 4
y = 9/4 - 18/4 - 16/4
y = -25/4
Therefore, the vertex is (3/2, -25/4).
In summary, the y-intercept is -4, the x-intercepts are 4 and -1, and the vertex is located at (3/2, -25/4).