Find the x and y Intercepts y=4x^2+19x+12

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: $(-\frac{3}{4},0),(-4,0)$.

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: $(-\frac{3}{4},0),(-4,0)$ y-intercept: $(0,12)$

Step:4

(No further action required in this step)

Knowledge Notes:

To find the x-intercepts of a quadratic equation $y=a{x}^2+bx+c$, one must set $y$ to $0$ and solve the resulting quadratic equation $a{x}^2+bx+c=0$. 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 $(0,y)$.