We want to solve the given quadratic equation by graphing.

x^{2} + 3 x - 10 = 0

To do so, we will graph the quadratic function represented by the left-hand side of the above equation. To draw the graph, we must start by identifying the values of

a,

b,

and

c .

y = x^{2} + 3 x - 10 \Leftrightarrow y = 1 x^{2} + 3 x + (- 10)

We can see that

a = 1,

b = 3,

and

c = - 10 .

Now, we will follow four steps to graph the function.

Find the axis of symmetry.
Calculate the vertex.
Identify the $y -$ intercept and its reflection across the axis of symmetry.
Connect the points with a parabola.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with equation

x = - \frac{b}{2 a} .

Since we already know the values of

a

and

b,

we can substitute them into the formula.

x = - \frac{b}{2 a}

SubstituteII

$a = 1$ , $b = 3$

x = - \frac{3}{2 ( 1 )}

MultByOne

$a \cdot 1 = a$

x = - \frac{3}{2}

CalcQuot

Calculate quotient

x = - 1.5

The axis of symmetry of the parabola is the vertical line with equation

x = - 1.5 .

Calculating the Vertex

To calculate the vertex, we need to think of

y

as a function of

x,

y = f (x) .

We can write the expression for the vertex by stating the

x -

and

y -

coordinates in terms of

a

and

b .

Vertex : (- \frac{b}{2 a}, f (- \frac{b}{2 a}))

Note that the formula for the

x -

coordinate is the same as the formula for the axis of symmetry, which is

x = - 1.5 .

Thus, the

x -

coordinate of the vertex is also

- 1.5 .

To find the

y -

coordinate, we need to substitute

- 1.5

for

x

in our function.

y = x^{2} + 3 x - 10

Substitute

$x = - 1.5$

y = (- 1.5)^{2} + 3 (- 1.5) - 10

▼

Simplify right-hand side

CalcPow

Calculate power

y = 2.25 + 3 (- 1.5) - 10

MultPosNeg

$a (- b) = - a \cdot b$

y = 2.25 - 4.5 - 10

SubTerms

Subtract terms

y = - 12.25

We found the

y -

coordinate, and now we know that the vertex is

(- 1.5, - 12.25) .

Identifying the $y -$ intercept and its Reflection

The $y -$ intercept of the graph of a quadratic function written in standard form is given by the value of $c .$ Thus, the point where our graph intercepts the $y -$ axis is $(0, - 10) .$ Let's plot this point and its reflection across the axis of symmetry.

Connecting the Points

We can now draw the graph of the function. Since $a = 1,$ which is positive, the parabola will open upward. Let's connect the three points with a smooth curve.

The $x -$ intercepts of the graph are the solutions to the given equation.

By looking at the graph, we can state the values for the $x -$ intercepts. We can see that the parabola intercepts the $x -$ axis at $(- 5, 0)$ and $(2, 0) .$ Therefore, the solutions to the equation are $x = - 5$ and $x = 2 .$

Finding the Axis of Symmetry

Calculating the Vertex

Identifying the y-intercept and its Reflection

Connecting the Points

Identifying the $y -$ intercept and its Reflection

Exercise