**In this method, we are going to find the solution or roots of a quadratic equation graphically. The graph of a quadratic equation is always a parabola. The points where the graph of the parabola cuts the**

*x-*axis are called the roots of the corresponding quadratic equation.**To graph a quadratic equation, we should graph the related quadratic function. For example, if the quadratic equation is**

*x*^{2}+ 2*x*+ 5 = 0, then the related quadratic function is*f*(*x*) =*x*^{2}+ 2*x*+ 5.**Let us solve few problems on quadratic equations by graphing.**

**Example 1:**

**Solve**

*x*^{2}+*x*- 6 = 0 by graphing.**Solution:**

**Step 1:**

**Given equation is**

*x*^{2}+*x*- 6 = 0.**Here**

*a*= 1,*b*= 1, and*c*= -6.**Graph the related function f(**

*x*) =*x*^{2}+*x*- 6.**The equation of the axis of symmetry is**

*x*= -^{b}/_{2a}

*x*= -^{b}/_{2a}= -^{1}/_{2(1) }= -^{1}/_{2}= -0.5**=>**

*x*= -0.5 is the axis of symmetry of the equation.**Step 2:**

**When**

*x*= -0.5,

*f*(*x*) =*f*(-0.5) = (-0.5)^{ 2}+ (-0.5) - 6 = 0**=>**

*f*(-0.5) = 0.25 - 0.5 - 6 = -6.25**So, the coordinates of the vertex are (-0.5, -6.25).**

**Step 3:**

**Now, make the table with all other points.**

**Put**

*x*= -4, -3, -2, 0, 1, 2, 3 and find f(*x*).**When**

*x*= -4,

*f*(*x*) =*f*(-4) = (-4)^{2}+ (-4) - 6 = 16 - 4 - 6 = 6**When**

*x*= -3,

*f*(*x*) =*f*(-3) = (-3)^{2}+ (-3) - 6 = 9 - 3 - 6 = 0**When**

*x*= -2,

*f*(*x*) =*f*(-2) = (-2)^{2}+ (-2) - 6 = 4 - 2 - 6 = -4**When**

*x*= -1,

*f*(*x*) =*f*(-1) = (-1)^{2}+ (-1) - 6 = 1 - 1 - 6 = -6**When**

*x*= 0,

*f*(*x*) =*f*(0) = (0)^{2}+ (0) - 6 = 0 + 0 - 6 = -6**When**

*x*= 1,

*f*(*x*) =*f*(1) = (1)^{2}+ (1) - 6 = 1 + 1 - 6 = -4**When**

*x*= 2,

*f*(*x*) =*f*(2) = (2)^{2}+ 2 - 6 = 4 + 2 - 6 = 0**When**

*x*= 3,

*f*(*x*) =*f*(3) = (3)^{2}+ 3 - 6 = 9 + 3 - 6 = 6x | -4 | -3 | -2 | -1 | -0.5 | 0 | 1 | 2 | 3 |

f(x) | 6 | 0 | -4 | -6 | -6.25 | -6 | -4 | 0 | 6 |

**Hence, the ordered pairs of the quadratic function are (-4, 6), (-3, 0), (-2, -4), (-1, -6), (-0.5, -6.25), (0, -6), (1, -4), (2, 0), and (3, 6).**

**Step 4:**

**Now, plot the ordered pairs in a coordinate plane and join the points.**

**The point where the related function**

*f*(*x*) = 0 is the solution to this quadratic equation. This occurs at the*x*-intercepts. The*x*-intercepts are the points where the graph touches the*x*-axis. So, the graph has -3 and 2 as*x*-intercepts.**So, the solutions are -3 and 2.**

**Step 5:**

**Check:**

**We can also check our solutions by factoring the given equation.**

*x*^{2}+*x*- 6 =0**=>**

*x*^{2}+ 3*x*- 2*x*- 6 = 0**=>**

*x*(*x*+ 3) - 2(*x*+ 3) = 0**=> (**

*x*+ 3)(*x*- 2) = 0**Using the zero product property,**

*x*+ 3 = 0 or*x*- 2 = 0

*x*= -3 or*x*= 2**Step 6:**

**So, the solutions of the equation are -3 and 2.**

**Example 2:**

**Solve**

*x*^{2}- 10*x*+ 25 = 0 by graphing.**Solution:**

**Step 1:**

**Given equation is**

*x*^{2}- 10*x*+ 25 = 0.**Here**

*a*= 1,*b*= -10, and*c*= 25.**Graph the related function f(**

*x*) =*x*^{2}- 10*x*+ 25.**The equation of the axis of symmetry is**

*x*= -^{b}/_{2a}

*x*= -^{b}/_{2a}= -^{10}/_{2(1)}= -^{b}/_{2a}= 5**=>**

*x*= 5 is the axis of symmetry of the given equation.**Step 2:**

**When**

*x*= 5,

*f*(*x*) =*f*(5) = (5)^{2}- 10(5) + 25 = 25 - 50 + 25 = 0**=>**

*f*(*x*) = 0**So, the coordinates of the vertex are (5, 0).**

**Step 3:**

**Now, make the table with all other points.**

**Put**

*x*= 2, 3, 4, 6, 7, 8 and find f(*x*).**When**

*x*= 2,

*f*(*x*) =*f*(2) = (2)^{2}- 10(2) + 25 = 4 - 20 + 25 = 9**When**

*x*= 3,

*f*(*x*) =*f*(3) = (3)^{2}- 10(3) + 25 = 9 - 30 + 25 = 4**When**

*x*= 4,

*f*(*x*) =*f*(4) = (4)^{2}- 10(4) + 25 = 16 - 40 + 25 = 1**When**

*x*= 6,

*f*(*x*) =*f*(6) = (6)^{2}- 10(6) + 25 = 36 - 60 + 25 = 1**When**

*x*= 7,

*f*(*x*) =*f*(7) = (7)^{2}- 10(7) + 25 = 49 - 70 + 25 = 4**When**

*x*= 8,

*f*(*x*) =*f*(8) = (8)^{2}- 10(8) + 25 = 64 - 80 + 25 = 9x | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

f(x) | 9 | 4 | 1 | 0 | 1 | 4 | 9 |

**So, the ordered pairs of the quadratic function are (2, 9), (3, 4), (4, 1), (5, 0), (6, 1), (7, 4), and (8, 9).**

**Step 4:**

**Now, plot the ordered pairs in a coordinate plane and join the points.**

**The resultant graph will be as shown.**

**The point where the related function**

*f*(*x*) = 0 is the solution to this quadratic equation. This occurs at the*x*-intercepts. The*x*-intercepts are the points where the graph touches the*x*-axis.**Notice that the vertex of the parabola is the**

*x*-intercept.**Hence, the solution is 5.**

**Step 5:**

**Check:**

**Let us check it by factoring.**

**Factor the equation**

*x*^{2}- 10*x*+ 25 = 0.

*x*^{2}- 5*x*- 5*x*+ 25 = 0**=>**

*x*(*x*- 5) - 5(*x*- 5) = 0**=> (**

*x*- 5)(*x*- 5) = 0**=>**

*x*- 5 = 0 or*x*- 5 = 0**=>**

*x*= 5 or*x*= 5**So, there are two roots but they are identical.**

**So, this quadratic equation has only one root called 5 as**

*double root*.**Step 6:**

**So, the solution is 5.**

**Example 3:**

**Solve**

*x*^{2}+ 2*x*+ 5 = 0 by graphing.**Solution:**

**Step 1:**

**Given equation is**

*x*^{2}+ 2*x*+ 5 = 0.**Here,**

*a*= 1,*b*= 2, and*c*= 5.**Graph the related function f(**

*x*) =*x*^{2}+ 2*x*+ 5.**The equation of the axis of symmetry is**

*x*= -^{b}/_{2a}

*x*= -^{b}/_{2a}= -^{2}/_{2(1)}= -1**=>**

*x*= -1 is the axis of symmetry of the given equation.**Step 2:**

**When**

*x*= -1,

*f*(*x*) =*f*(-1) = (-1)^{2}+ 2(-1) + 5 = 1 - 2 + 5 = 4**So, the coordinates of the vertex are (-1, 4).**

**Now, make the table with all other points.**

**Put**

*x*= -4, -3, -2, 0, 1, 2 and find f(*x*).**When**

*x*= -4,

*f*(*x*) =*f*(-4) = (-4)^{2}+ 2(-4) + 5 = 16 - 8 + 5 = 13**=>**

*f*(-4) = 13**When**

*x*= -3,

*f*(*x*) =*f*(-3) = (-3)^{2}+ 2(-3) + 5 = 9 - 6 + 5 = 8**=>**

*f*(-3) = 8**When**

*x*= -2,

*f*(*x*) =*f*(-2) = (-2)^{2}+ 2(-2) + 5 = 4 - 4 +5 = 5**=>**

*f*(-2) = 5**When**

*x*= 0,

*f*(*x*) =*f*(0) = (0)^{2}+ 2(0) + 5 = 5**=>**

*f*(0) = 5**When**

*x*= 1,

*f*(*x*) =*f*(1) = (1)^{2}+ 2(1) + 5 = 1 + 2 + 5 = 8**=>**

*f*(1) = 8**When**

*x*= 2,

*f*(*x*) =*f*(2) = (2)^{2}+ 2(2) + 5 = 13**=>**

*f*(2) = 13x | -4 | -3 | -2 | -1 | 0 | 1 | 2 |

f(x) | 13 | 8 | 5 | 4 | 5 | 8 | 13 |

**So, the ordered pairs of the quadratic function are (-4, 13), (-3, 8), (-2, 5), (-1, 4), (0, 5), (1, 8), and (2, 13).**

**Step 3:**

**Now, plot the ordered pairs in a coordinate plane and join the points.**

**The resultant graph will be as shown.**

**The point where the related function**

*f*(*x*) = 0 is the solution to this quadratic equation. This occurs at the*x*-intercepts. The*x*-intercepts are the points where the graph touches the*x*-axis. So, the graph has no*x*-intercept.**Step 4:**

**So, there are no real number solutions for the given quadratic equation.**

**Here is some practice questions:**

**Quadratic equations**

**To check your result:**

**Quadratic equation solver**

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