The given quadratic equation is,ax2 + bx + c = 0Now, sin θ and cos θ are the roots of the above equation Now, sin θ + cos θ = -baNow, sin θ cos θ = caNow, sin θ + cos θ2 = sin2θ + cos2θ + 2 sin θ cos θ⇒-ba2 = 1 + 2 × ca⇒b2a2 = 1 + 2ca⇒b2a2 = a + 2ca⇒b2a = a + 2c⇒a2 + 2ac = b2⇒a2 - b2 + 2ac = 0

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The given quadratic equation is, {ax}^{2} + bx + c = 0 Now, \sin θ and \cos θ are the roots of the above equation . Now, \sin θ + \cos θ = - \frac{b}{a} Now, \sin θ . \cos θ = \frac{c}{a} Now, {(\sin θ + \cos θ)}^{2} = \sin^{2} θ + \cos^{2} θ + 2 \sin θ . \cos θ \Rightarrow {(- \frac{b}{a})}^{2} = 1 + 2 \times \frac{c}{a} \Rightarrow \frac{b^{2}}{a^{2}} = 1 + \frac{2 c}{a} \Rightarrow \frac{b^{2}}{a^{2}} = \frac{a + 2 c}{a} \Rightarrow \frac{b^{2}}{a} = a + 2 c \Rightarrow a^{2} + 2 ac = b^{2} \Rightarrow a^{2} - b^{2} + 2 ac = 0

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