Find the x and y Intercepts y=4x^2+19x+12
The problem asks to identify the points at which the given quadratic function crosses the x-axis and the y-axis. The x-intercepts are the values of 'x' for which 'y' equals zero, and the y-intercept is the value of 'y' when 'x' equals zero. Specifically, the question requires solving for the roots (x-intercepts) of the quadratic equation when y=0, and calculating the function's value (y-intercept) when x=0.
$y = 4 x^{2} + 19 x + 12$
Answer
Solution:
Step:1
Determine the x-intercepts.
Step:1.1
Set $y$ to $0$ and solve for $x$: $0 = 4x^2 + 19x + 12$.
Step:1.2
Proceed to solve the quadratic equation.
Step:1.2.1
Rewrite the equation as $4x^2 + 19x + 12 = 0$.
Step:1.2.2
Employ the method of factorization.
Step:1.2.2.1
Split the middle term into two terms that multiply to $4 \cdot 12 = 48$ and add up to $19$.
Step:1.2.2.1.1
Extract $19x$ from $4x^2 + 19x + 12 = 0$.
Step:1.2.2.1.2
Decompose $19$ into $3 + 16$.
Step:1.2.2.1.3
Apply the distributive property to get $4x^2 + 3x + 16x + 12 = 0$.
Step:1.2.2.2
Factor out the common factor from each binomial.
Step:1.2.2.2.1
Pair the terms as $(4x^2 + 3x) + (16x + 12) = 0$.
Step:1.2.2.2.2
Extract the GCF from each pair to get $x(4x + 3) + 4(4x + 3) = 0$.
Step:1.2.2.3
Factor by taking out the common binomial factor $(4x + 3)(x + 4) = 0$.
Step:1.2.3
Recognize that if any factor equals $0$, the entire expression is $0$.
Step:1.2.4
Isolate $4x + 3 = 0$ and solve for $x$.
Step:1.2.4.1
Set $4x + 3$ to $0$.
Step:1.2.4.2
Resolve the equation $4x + 3 = 0$ for $x$.
Step:1.2.4.2.1
Subtract $3$ from both sides to get $4x = -3$.
Step:1.2.4.2.2
Divide by $4$ to isolate $x$.
Step:1.2.4.2.2.1
Divide $4x$ and $-3$ by $4$: $\frac{4x}{4} = \frac{-3}{4}$.
Step:1.2.4.2.2.2
Simplify to find $x = \frac{-3}{4}$.
Step:1.2.5
Solve $x + 4 = 0$ for $x$.
Step:1.2.5.1
Set $x + 4$ to $0$.
Step:1.2.5.2
Subtract $4$ to find $x = -4$.
Step:1.2.6
The solutions are the x-values for which $(4x + 3)(x + 4) = 0$.
Step:1.3
Express the x-intercepts as points: $\left(-\frac{3}{4}, 0\right), (-4, 0)$.
Step:2
Identify the y-intercept.
Step:2.1
Set $x$ to $0$ and solve for $y$: $y = 4(0)^2 + 19(0) + 12$.
Step:2.2
Simplify the equation.
Step:2.2.1
Eliminate parentheses to get $y = 4 \cdot 0^2 + 19 \cdot 0 + 12$.
Step:2.2.2
Recognize that any term multiplied by $0$ is $0$.
Step:2.2.3
Simplify to find $y = 12$.
Step:2.3
State the y-intercept as a point: $(0, 12)$.
Step:3
Compile the list of intercepts.
x-intercepts: $\left(-\frac{3}{4}, 0\right), (-4, 0)$ y-intercept: $(0, 12)$
Step:4
(No further action required in this step)
Knowledge Notes:
To find the x-intercepts of a quadratic equation $y = ax^2 + bx + c$, one must set $y$ to $0$ and solve the resulting quadratic equation $ax^2 + bx + c = 0$. This can be done by factoring, completing the square, or using the quadratic formula.
The y-intercept is found by setting $x$ to $0$ and solving for $y$. This is typically straightforward as it involves evaluating the constant term of the quadratic equation.
When factoring a quadratic equation, one approach is to use factor by grouping. This involves finding two numbers that multiply to give $ac$ (the product of the coefficients of $x^2$ and the constant term) and add up to $b$ (the coefficient of $x$). These two numbers are then used to split the middle term and factor by grouping.
The x-intercepts are the points where the graph of the equation crosses the x-axis, and they have coordinates of the form $(x, 0)$. The y-intercept is the point where the graph crosses the y-axis and has coordinates of the form $(0, y)$.
