Find the x and y Intercepts y=-6x^2-5x-1

Solution:

Step:1

Determine the x-intercepts.

Step:1.1

Set $y$ to $0$ and solve for $x$: $0=-6x^2-5x-1$.

Step:1.2

Proceed to solve the quadratic equation.

Step:1.2.1

Rewrite the equation as $-6x^2-5x-1=0$.

Step:1.2.2

Begin factoring the quadratic expression.

Step:1.2.2.1

Extract $-1$ from each term: $-(6x^2+5x+1)=0$.

Step:1.2.2.2

Attempt to factor by grouping.

Step:1.2.2.2.1

Split the middle term to factor by grouping, ensuring the two terms multiply to $6\cdot 1=6$ and add up to $5$.

Step:1.2.2.2.2

Factor out the common factor from each binomial: $-(2x(3x+1)+1(3x+1))=0$.

Step:1.2.2.2.3

Factor out the common binomial factor: $-(3x+1)(2x+1)=0$.

Step:1.2.3

Set each factor equal to $0$: $3x+1=0$ and $2x+1=0$.

Step:1.2.4

Solve $3x+1=0$ for $x$: $x=-\frac{1}{3}$.

Step:1.2.5

Solve $2x+1=0$ for $x$: $x=-\frac{1}{2}$.

Step:1.2.6

The solutions are $x=-\frac{1}{3}$ and $x=-\frac{1}{2}$.

Step:1.3

Express the x-intercepts as ordered pairs: $(-\frac{1}{3},0)$ and $(-\frac{1}{2},0)$.

Step:2

Identify the y-intercepts.

Step:2.1

Set $x$ to $0$ and solve for $y$: $y=-6(0{)}^2-5\cdot 0-1$.

Step:2.2

Simplify the equation to find $y$: $y=-1$.

Step:2.3

Express the y-intercept as an ordered pair: $(0,-1)$.

Step:3

Combine the intercepts.

List the x-intercepts and y-intercept: $(-\frac{1}{3},0)$, $(-\frac{1}{2},0)$, and $(0,-1)$.

Knowledge Notes:

To find the x-intercepts of a function, we set $y$ to $0$ and solve the resulting equation for $x$. The x-intercepts are the points where the graph of the function crosses the x-axis.

To find the y-intercepts, we set $x$ to $0$ and solve for $y$. The y-intercepts are the points where the graph of the function crosses the y-axis.

The quadratic equation $a{x}^2+bx+c=0$ can be solved by factoring, completing the square, or using the quadratic formula. In this case, we attempted to factor the equation.

Factoring by grouping is a method used when a polynomial does not factor easily. It involves rearranging and grouping terms to find common factors.

The distributive property, $a(b+c)=ab+ac$, is used in the factoring process.

The solutions to the quadratic equation are the values of $x$ that make the equation true, and these correspond to the x-intercepts of the graph.

Ordered pairs are used to represent points on a coordinate plane, with the x-coordinate first and the y-coordinate second, in the form $(x,y)$.