Question

1. $6p^2 - 2p - 3 = 0$
2. $-2x^2 - x - 1 = 0$
3. $-4m^2 - 4m + 5 = 0$
4. $5b^2 + b - 2 = 0$
5. $r^2 + 5r + 2 = 0$
6. $2p^2 + 5p - 4 = 0$

Explanation:

Let's solve equation 1: \(6p^{2}-2p - 3=0\) using the quadratic formula. For a quadratic equation \(ax^{2}+bx + c = 0\), the solutions are \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a = 6\), \(b=-2\), \(c=-3\).

Step 1: Calculate the discriminant \(\Delta=b^{2}-4ac\)

Substitute \(a = 6\), \(b=-2\), \(c=-3\) into the discriminant formula:
\(\Delta=(-2)^{2}-4\times6\times(-3)\)
\(=4 + 72\)
\(=76\)

Step 2: Find the roots using the quadratic formula

\(p=\frac{-(-2)\pm\sqrt{76}}{2\times6}=\frac{2\pm2\sqrt{19}}{12}=\frac{1\pm\sqrt{19}}{6}\)

Now solve equation 2: \(-2x^{2}-x - 1=0\) or \(2x^{2}+x + 1=0\) (multiplying both sides by - 1). Here, \(a = 2\), \(b = 1\), \(c = 1\)

Step 3: Calculate the discriminant \(\Delta=b^{2}-4ac\)

\(\Delta=1^{2}-4\times2\times1=1 - 8=-7<0\), so no real roots.

Equation 3: \(-4m^{2}-4m + 5=0\) or \(4m^{2}+4m - 5=0\) (multiplying by - 1). \(a = 4\), \(b = 4\), \(c=-5\)

Step 4: Calculate the discriminant \(\Delta=b^{2}-4ac\)

\(\Delta=4^{2}-4\times4\times(-5)=16 + 80=96\)

Step 5: Find the roots

\(m=\frac{-4\pm\sqrt{96}}{2\times4}=\frac{-4\pm4\sqrt{6}}{8}=\frac{-1\pm\sqrt{6}}{2}\)

Equation 4: \(5b^{2}+b - 2=0\), \(a = 5\), \(b = 1\), \(c=-2\)

Step 6: Calculate the discriminant \(\Delta=b^{2}-4ac\)

\(\Delta=1^{2}-4\times5\times(-2)=1 + 40=41\)

Step 7: Find the roots

\(b=\frac{-1\pm\sqrt{41}}{2\times5}=\frac{-1\pm\sqrt{41}}{10}\)

Equation 5: \(r^{2}+5r + 2=0\), \(a = 1\), \(b = 5\), \(c = 2\)

Step 8: Calculate the discriminant \(\Delta=b^{2}-4ac\)

\(\Delta=5^{2}-4\times1\times2=25 - 8=17\)

Step 9: Find the roots

\(r=\frac{-5\pm\sqrt{17}}{2\times1}=\frac{-5\pm\sqrt{17}}{2}\)

Equation 6: \(2p^{2}+5p - 4=0\), \(a = 2\), \(b = 5\), \(c=-4\)

Step 10: Calculate the discriminant \(\Delta=b^{2}-4ac\)

\(\Delta=5^{2}-4\times2\times(-4)=25 + 32=57\)

Step 11: Find the roots

\(p=\frac{-5\pm\sqrt{57}}{2\times2}=\frac{-5\pm\sqrt{57}}{4}\)

Answer:

1. \(p=\frac{1\pm\sqrt{19}}{6}\)
2. No real roots
3. \(m=\frac{-1\pm\sqrt{6}}{2}\)
4. \(b=\frac{-1\pm\sqrt{41}}{10}\)
5. \(r=\frac{-5\pm\sqrt{17}}{2}\)
6. \(p=\frac{-5\pm\sqrt{57}}{4}\)