Quadratic equations: Quadratic inequalities
Quadratic inequalities
Just as with linear inequalities, we can create an inequality with quadratics. We will first take a look at how to solve a quadratic inequality.
We solve the inequality #\blue{2x^2+4x+3} \gt \green{x^22x+6}#.
Step 1 
We first solve the equality #\blue{2x^2+4x+3} = \green{x^22x+6}#. Therefore we first reduce the equality to #0#. \[\begin{array}{rcl}2x^2+4x+3&=&x^22x+6 \\ &&\phantom{xx}\blue{\text{the equation to solve}}\\ 3x^2+6x3 &=&0 \\ &&\phantom{xx}\blue{\text{reduce to }0} \end{array}\] Next we apply the quadratic formula. Therefore we define the letters #a#, #b# and #c#. \[a=3, b=6 \text{ and } c=3\] Next, we calculate the discriminant. \[D=6^24 \cdot 3 \cdot 3 =72\] After that, we determine the solutions \[x=\frac{6\sqrt{72}}{2\cdot 3} \lor x=\frac{6+\sqrt{72}}{2\cdot 3} \] Which we simplify to: \[x=1 \sqrt{2} \lor x=1+ \sqrt{2} \] 
Step 2 
We create the graphs of #y=\blue{2x^2+4x+3}# and #y=\green{x^22x+6}\ (dashed)#. The intersection points are drawn in orange. 
Step 3 
We will determine the solution by means of step 1 and 2. On the left of #x=1\sqrt{2}# and on the right of #x=1+\sqrt{2}# the graph of #y=\blue{2x^2+4x+3}# lies above the graph of #y=\green{x^22x+6}#. hence, the solution is #x \lt 1\sqrt{2} \lor x \gt 1+\sqrt{2}#. 
In general, we can apply the following procedure.
Solving a quadratic inequality
Procedure  
We solve the inequality #\blue{a_1 x^2+b_1x+c_1} \gt \green{a_2x^2+b_2x+c_2}\ (dashed)#, in which #a_1 \ne 0#. 


Step 1  We first solve the equality \[\blue{a_1 x^2+b_1x+c_1} = \green{a_2x^2+b_2x+c_2}\]  
Step 2  We draw the graphs #y=\blue{a_1 x^2+b_1x+c_1}# and #y=\green{a_2x^2+b_2x+c_2}#.  
Step 3  With help of step 1 and 2, determine for which value of #x# the inequality holds. In the coordinate system, the bigger graph is the one that lies above the other. 
Note that this procedure also holds for the inequality signs #\geq#, but the #x#values of the intersection points are also part of the solution.
Step 1  We solve the equality #6\cdot q^2q+1=2\cdot q^24\cdot q+7#. This is done in the following way: \[\begin{array}{rcl} 6\cdot q^2q+1 &=& 2\cdot q^24\cdot q+7\\ &&\phantom{xxx}\blue{\text{the original equality}}\\ 4\cdot q^2+3\cdot q6&=&0\\ &&\phantom{xxx}\blue{\text{all terms moved to the left hand side}}\\ \text{discriminant } &=& b^24ac \\ &&\phantom{xxx}\blue{\text{formula discriminant}}\\ &=& 3^2  4 \cdot 4 \cdot \left(6\right) \\&&\phantom{xxx}\blue{\text{formula entered}}\\ &=& 105\\&&\phantom{xxx}\blue{\text{calculated}}\\ \text{number of solutions } &=& 2\\&&\phantom{xxx}\blue{\text{since discrimant bigger than }0}\\ q=\dfrac{b\sqrt{D}}{2 \cdot a} &\lor& q=\dfrac{b+\sqrt{D}}{2 \cdot a} \\&&\phantom{xxx}\blue{\text{formula solutions}}\\ q=\dfrac{3  \sqrt{105}}{2 \cdot 4} &\lor& q=\dfrac{3 + \sqrt{105}}{2 \cdot 4} \\&&\phantom{xxx}\blue{\text{formula entered}}\\ \displaystyle q= {{\sqrt{105}3}\over{8}} &\lor& \displaystyle q = {{\sqrt{105}3}\over{8}} \\&&\phantom{xxx}\blue{\text{calculated}}\\ \end{array}\] 
Step 2  We sketch the graphs of #y=6\cdot q^2q+1# (blue solid) and #y=2\cdot q^24\cdot q+7# (green dashed). 
Step 3  We can now read the answer of the inequality. \[q\gt {{\sqrt{105}3}\over{8}}\land q\lt {{\sqrt{105}3}\over{8}}\] 
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